Most contemporary generative models of images, sound and video do not operate directly on pixels or waveforms. They consist of two stages: first, a compact, higher-level latent representation is extracted, and then an iterative generative process operates on this representation instead. How does this work, and why is this approach so popular?

Generative models that make use of latent representations are everywhere nowadays, so I thought it was high time to dedicate a blog post to them. In what follows, I will talk at length about latents as a plural noun, which is the usual shorthand for latent representation. This terminology originated in the concept of latent variables in statistics, but it is worth noting that the meaning has drifted somewhat in this context. These latents do not represent any known underlying physical quantity which we cannot measure directly; rather, they capture perceptually meaningful information in a compact way, and in many cases they are a deterministic nonlinear function of the input signal (i.e. not random variables).

In what follows, I will assume a basic understanding of neural networks, generative models and related concepts. Below is an overview of the different sections of this post. Click to jump directly to a particular section.

1. The recipe
2. How we got here
3. Why two stages?
4. Trading off reconstruction quality and modelability
5. Controlling capacity
6. Curating and shaping the latent space
7. The tyranny of the grid
8. Latents for other modalities
9. Will end-to-end win in the end?
10. Closing thoughts
11. Acknowledgements
12. References

The recipe

The usual process for training a generative model in latent space consists of two stages:

1. Train an autoencoder on the input signals. This is a neural network consisting of two subnetworks, an encoder and a decoder. The former maps an input signal to its corresponding latent representation (encoding). The latter maps the latent representation back to the input domain (decoding).
2. Train a generative model on the latent representations. This involves taking the encoder from the first stage, and using it to extract latents for the training data. The generative model is then trained directly on these latents. Nowadays, this is usually either an autoregressive model or a diffusion model.

Once the autoencoder is trained in the first stage, its parameters will not change any further in the second stage: gradients from the second stage of the learning process are not backpropagated into the encoder. Another way to say this is that the encoder parameters are frozen in the second stage.

Note that the decoder part of the autoencoder plays no role in the second stage of training, but we will need it when sampling from the generative model, as that will generate outputs in latent space. The decoder enables us to map the generated latents back to the original input space.

Below is a diagram illustrating this two-stage training recipe. Networks whose parameters are learnt in the respective stages are indicated with a \(\nabla\) symbol, because this is almost always done using gradient-based learning. Networks whose parameters are frozen are indicated with a snowflake.

Several different loss functions are involved in the two training stages, which are indicated in red on the diagram:

• To ensure the encoder and decoder are able to convert input representations to latents and back with high fidelity, several loss functions constrain the reconstruction (decoder output) with respect to the input. These usualy include a simple regression loss, a perceptual loss and an adversarial loss.
• To constrain the capacity of the latents, an additional loss function is often applied directly to them during training, although this is not always the case. We will refer to this as the bottleneck loss, because the latent representation forms a bottleneck in the autoencoder network.
• In the second stage, the generative model is trained using its own loss function, separate from those used during the first stage. This is often the negative log-likelihood loss (for autoregressive models), or a diffusion loss.

Taking a closer look at the reconstruction-based losses, we have:

• the regression loss, which is sometimes the mean absolute error (MAE) measured in the input space (e.g. on pixels), but more often the mean squared error (MSE).
• the perceptual loss, which can take many forms, but more often than not, it makes use of another frozen pre-trained neural network to extract perceptual features. The loss encourages these features to match between the reconstruction and the input, which results in better preservation of high-frequency content that is largely ignored by the regression loss. LPIPS¹ is a popular choice for images.
• the adversarial loss, which uses a discriminator network which is co-trained with the autoencoder, as in generative adversarial networks (GANs)². The discriminator is trained to tell apart real input signals from reconstructions, and the autoencoder is trained to fool the discriminator into making mistakes. The goal is to improve the realism of the output, even if it means deviating further from the input signal. It is quite common for the adversarial loss to be disabled for some time at the start of training, to avoid instability.

Below is a more elaborate diagram of the first training stage, explicitly showing the other networks which typically play a role in this process.

It goes without saying that this generic recipe is often deviated from in one or more ways, especially for audio and video, but I have tried to summarise the most common ingredients found in most modern practical applications of this modelling approach.

How we got here

The two dominant generative modelling paradigms of today, autoregression and diffusion, were both initially applied to “raw” digital representations of perceptual signals, by which I mean pixels and waveforms. PixelRNN³ and PixelCNN⁴ generated images one pixel at a time. WaveNet⁵ and SampleRNN⁶ did the same for audio, producing waveform amplitudes one sample at a time. On the diffusion side, the original works that introduced⁷ and established^{8 9} the modelling paradigm all operated on pixels to produce images, and early works like WaveGrad¹⁰ and DiffWave¹¹ generated waveforms to produce sound.

However, it became clear very quickly that this strategy makes scaling up quite challenging. The most important reason for this can be summarised as follows: perceptual signals mostly consist of imperceptible noise. Or, to put it a different way: out of the total information content of a given signal, only a small fraction actually affects our perception of it. Therefore, it pays to ensure that our generative model can use its capacity efficiently, and focus on modelling just that fraction. That way, we can use smaller, faster and cheaper generative models without compromising on perceptual quality.

Latent autoregression

Autoregressive models of images took a huge leap forward with the seminal VQ-VAE paper¹². It suggested a practical strategy for learning discrete representations with neural networks, by inserting a vector quantisation bottleneck layer into an autoencoder. To learn such discrete latents for images, a convolutional encoder with several downsampling stages produced a spatial grid of vectors with a 4× lower resolution than the input (along both height and width, so 16× fewer spatial positions), and these vectors were then quantised by the bottleneck layer.

Now, we could generate images with PixelCNN-style models one latent vector at a time, rather than having to do it pixel by pixel. This significantly reduced the number of autoregressive sampling steps required, but perhaps more importantly, measuring the likelihood loss in the latent space rather than pixel space helped avoid wasting capacity on imperceptible noise. This is effectively a different loss function, putting more weight on perceptually relevant signal content, because a lot of the perceptually irrelevant signal content is not present in the latent vectors (see my blog post on typicality for more on this topic). The paper showed 128×128 generated images from a model trained on ImageNet, a resolution that had only been attainable with GANs² up to that point.

The discretisation was critical to its success, because autoregressive models were known to work much better with discrete inputs at the time. But perhaps even more importantly, the spatial structure of the latents allowed existing pixel-based models to be adapted very easily. Before this, VAEs (variational autoencoders^{13 14}) would typically compress an entire image into a single latent vector, resulting in a representation without any kind of topological structure. The grid structure of modern latent representations, which mirrors that of the “raw” input representation, is exploited in the network architecture of generative models to increase efficiency (through e.g. convolutions, recurrence or attention layers).

VQ-VAE 2¹⁵ further increased the resolution to 256x256 and dramatically improved image quality through scale, as well as the use of multiple levels of latent grids, structured in a hierarchy. This was followed by VQGAN¹⁶, which combined the adversarial learning mechanism of GANs with the VQ-VAE architecture. This enabled a dramatic increase of the resolution reduction factor from 4× to 16× (256× fewer spatial positions compared to pixel input), while still allowing for sharp and realistic reconstructions. The adversarial loss played a big role in this, encouraging realistic decoder output even when it is not possible to closely adhere to the original input signal.

VQGAN became a core technology enabling the rapid progress in generative modelling of perceptual signals that we’ve witnessed in the last five years. Its impact cannot be overestimated – I’ve gone as far as to say that it’s probably the main reason why GANs deserved to win the Test of Time award at NeurIPS 2024. The “assist” that the VQGAN paper provided, kept GANs relevant even after they were all but replaced by diffusion models for the base task of media generation.

IMO VQGAN is why GANs deserve the NeurIPS test of time award. Suddenly our image representations were an order of magnitude more compact. Absolute game changer for generative modelling at scale, and the basis for latent diffusion models.https://t.co/Ochh17IvGx

— Sander Dieleman (@sedielem) November 28, 2024

It is also worth pointing out just how much of the recipe from the previous section was conceived in this paper. The iterative generator isn’t usually autoregressive these days (Parti¹⁷, xAI’s recent Aurora model and, apparently, OpenAI’s GPT-4o are notable exceptions), and the quantisation bottleneck has been replaced, but everything else is still there. Especially the combination of a simple regression loss, a perceptual loss and an adversarial loss has stubbornly persisted, in spite of its apparent complexity. This kind of endurance is rare in a fast-moving field like machine learning – perhaps rivalled only by that of the largely unchanged Transformer architecture¹⁸ and the Adam optimiser¹⁹!

(While discrete representations played an essential role in making latent autoregression work at scale, I wanted to point out that autoregression in continuous space has also been made to work well recently^{20 21}.)

Latent diffusion

With latent autoregression gaining ground in the late 2010s, and diffusion models breaking through in the early 2020s, combining the strengths of both approaches was a natural next step. As with many ideas whose time has come, we saw a string of concurrent papers exploring this topic hit arXiv around the same time, in the second half of 2021^{22 23 24 25 26}. The most well-known of these is Rombach et al.’s High-Resolution Image Synthesis with Latent Diffusion Models²⁶, who reused their previous VQGAN work¹⁶ and swapped out the autoregressive Transformer for a UNet-based diffusion model. This formed the basis for the Stable Diffusion models. Other works explored similar ideas, albeit at a smaller scale²⁴, or for modalities other than images²².

It took a little bit of time for the approach to become mainstream. Early commercial text-to-image models made use of so-called resolution cascades, consisting of a base diffusion model that generates low-resolution images directly in pixel space, and one or more upsampling diffusion models that produce higher-resolution outputs conditioned on lower-resolution inputs. Examples include DALL-E 2 and Imagen 2. After Stable Diffusion, most moved to a latent-based approach (including DALL-E 3 and Imagen 3).

An important difference between autoregressive and diffusion models is the loss function used to train them. In the autoregressive case, things are relatively simple: you just maximise the likelihood (although other things have been tried as well²⁷). For diffusion, things are a little more interesting: the loss is an expectation over all noise levels, and the relative weighting of these noise levels significantly affects what the model learns (for an explanation of this, see my previous blog post on noise schedules, as well as my blog post about casting diffusion as autoregression in frequency space). This justifies an interpretation of the typical diffusion loss as a kind of perceptual loss function, which puts more emphasis on signal content that is more perceptually salient.

At first glance, this makes the two-stage approach seem redundant, as it operates in a similar way, i.e. filtering out perceptually irrelevant signal content, to avoid wasting model capacity on it. If we can rely on the diffusion loss to focus only on what matters perceptually, why do we need a separate representation learning stage to filter out the stuff that doesn’t? These two mechanisms turn out to be quite complementary in practice however, for two reasons:

• The way perception works at small and large scales, especially in the visual domain, seems to be fundamentally different – to the extent that modelling texture and fine-grained detail merits separate treatment, and an adversarial approach can be more suitable for this. I will discuss this in more detail in the next section.
• Training large, powerful diffusion models is inherently computationally intensive, and operating in a more compact latent space allows us to avoid having to work with bulky input representations. This helps to reduce memory requirements and speeds up training and sampling.

Some early works did consider an end-to-end approach, jointly learning the latent representation and the diffusion prior^{23 25}, but this didn’t really catch on. Although avoiding sequential dependencies between multiple stages of training is desirable from a practical perspective, the perceptual and computational benefits make it worth the hassle.

Why two stages?

As discussed before, it is important to ensure that generative models of perceptual signals can use their capacity efficiently, as this makes them much more cost-effective. This is essentially what the two-stage approach accomplishes: by extracting a more compact representation that focuses on the perceptually relevant fraction of signal content, and modelling that instead of the original representation, we are able to make relatively modestly sized generative models punch above their weight.

The fact that most bits of information in perceptual signals don’t actually matter perceptually is hardly a new observation: it is also the key idea underlying lossy compression, which enables us to store and transmit these signals at a fraction of the cost. Compression algorithms like JPEG and MP3 exploit the redundancies present in signals, as well as the fact that our audiovisual senses are more sensitive to low frequencies than to high frequencies, to represent perceptual signals with far fewer bits. (There are other perceptual effects that play a role, such as auditory masking for example, but non-uniform frequency sensitivity is the most important one.)

So why don’t we use these lossy compression techniques as a basis for our generative models then? This is not a bad idea, and several works have used these algorithms, or parts of them, for this purpose^{28 29 30}. But a very natural reflex for people working on generative models is to try to solve the problem with more machine learning, to see if we can do better than these “handcrafted” algorithms.

It’s not just hubris on the part of ML researchers, though: there is actually a very good reason to use learned latents, instead of using these pre-existing compressed representations. Unlike in the compression setting, where smaller is better, and size is all that matters, the goal of generative modelling also imposes other constraints: some representations are easier to model than others. It is crucial that some structure remains in the representation, which we can then exploit by endowing the generative model with the appropriate inductive biases. This requirement creates a trade-off between reconstruction quality and modelability of the latents, which we will investigate in the next section.

An important reason behind the efficacy of latent representations is how they lean in to the fact that our perception works differently at different scales. In the audio domain, this is readily apparent: very rapid changes in amplitude result in the perception of pitch, whereas changes on coarser time scales (e.g. drum beats) can be individually discerned. Less well-known is that the same phenomenon also plays an important role in visual perception: rapid local fluctuations in colour and intensity are perceived as textures. A while back, I tried to explain this on Twitter, and I will paraphrase that explanation here:

Because of the dramatic improvements in efficiency that the two-stage approach offers, we seem to be happy to put up with the additional complexity it entails – at least, for now. This increased efficiency results in faster and cheaper training runs, but perhaps more importantly, it can greatly accelerate sampling as well. With generative models that perform iterative refinement, this significant cost reduction is of course very welcome, because many forward passes through the model are required to produce a single sample.

Trading off reconstruction quality and modelability

The difference between lossy compression and latent representation learning is worth exploring in more detail. One can use machine learning for both, although most lossy compression algorithms in widespread use today do not. These algorithms are typically rooted in rate-distortion theory, which formalises and quantifies the relationship between the degree to which we are able to compress a signal (rate), and how much we allow the decompressed signal to deviate from the original (distortion).

For latent representation learning, we can extend this trade-off by introducing the concept of modelability or learnability, which characterises how challenging it is for generative models to capture the distribution of this representation. This results in a three-way rate-distortion-modelability trade-off, which is closely related to the rate-distortion-usefulness trade-off discussed by Tschannen et al. in the context of representation learning³¹. (Another popular way to extend this trade-off in a machine learning context is the rate-distortion-perception trade-off³², which explicitly distinguishes reconstruction fidelity from perceptual quality. To avoid overcomplicating things, I will not make this distinction here, instead treating distortion as a quantity measured in a perceptual space, rather than input space.)

It’s not immediately obvious why this is even a trade-off at all – why is modelability at odds with distortion? To understand this, consider how lossy compression algorithms operate: they take advantage of known signal structure to reduce redundancy. In the process, this structure is often removed from the compressed representation, because the decompression algorithm is able to reconstitute it. But structure in input signals is also exploited extensively in modern generative models, in the form of architectural inductive biases for example, which take advantage of signal properties like translation equivariance or specific characteristics of the frequency spectrum.

If we have an amazing algorithm that efficiently removes almost all redundancies from our input signals, we are making it very difficult for generative models to capture the unstructured variability that remains in the compressed signals. That is completely fine if compression is all we are after, but not if we want to do generative modelling. So we have to strike a balance: a good latent representation learning algorithm will detect and remove some redundancy, but keep some signal structure as well, so there is something left for the generative model to work with.

A good example of what not to do in this setting is entropy coding, which is actually a lossless compression method, but is also used as the final stage in many lossy schemes (e.g. Huffman coding in JPEG/PNG, or arithmetic coding in H.265). Entropy coding algorithms reduce redundancy by assigning shorter representations to frequently occurring patterns. This doesn’t remove any information at all, but it destroys structure. As a result, small changes in input signals could lead to much larger changes in the corresponding compressed signals, potentially making entropy-coded sequences considerably more difficult to model.

In contrast, latent representations tend to preserve a lot of signal structure. The figure below shows a few visualisations of Stable Diffusion latents for images (taken from the EQ-VAE paper³³). It is pretty easy to identify the animals just from visually inspecting the latents. They basically look like noisy, low-resolution images with distorted colours. This is why I like to think of image latents as merely “advanced pixels”, capturing a little bit of extra information that regular pixels wouldn’t, but mostly still behaving like pixels nonetheless.

It is safe to say that these latents are quite low-level. Whereas traditional VAEs would compress an entire image into a single feature vector, often resulting in a high-level representation that enables semantic manipulation³⁴, modern latent representations used for generative modelling of images are actually much closer to the pixel level. They are much higher-capacity, inheriting the grid structure of the input (though at a lower resolution). Each latent vector in the grid may abstract away some low-level image features such as textures, but it does not capture the semantics of the image content. This is also why most autoencoders do not make use of any additional conditioning signals such as text captions, as those mainly constrain high-level structure (though exceptions exist³⁵).

Controlling capacity

Two key design parameters control the capacity of a grid-structured latent space: the downsampling factor and the number of channels of the representation. If the latent representation is discrete, the codebook size is also important, as it imposes a hard limit on the number of bits information that the latents can contain. (Aside from these, regularisation strategies play an important role, but we will discuss their impact in the next section.)

As an example, an encoder might take a 256×256 pixel image as input, and produce a 32×32 grid of continuous latent vectors with 8 channels. This could be achieved using a stack of strided convolutions, or perhaps using a vision Transformer (ViT)³⁶ with patch size 8. The downsampling factor reduces the dimensionality along both width and height, so there are 64 times fewer latent vectors than pixels – but each latent vector has 8 components, while each pixel has only 3 (RGB). In aggregate, the latent representation is a tensor with \(\frac{w_{in} \cdot h_{in} \cdot c_{in}}{w_{out} \cdot h_{out} \cdot c_{out}} = \frac{256 \cdot 256 \cdot 3}{32 \cdot 32 \cdot 8} = 24\) times fewer components (i.e. floating point numbers) than the tensor representing the original image. I like to refer to this number as the tensor size reduction factor (TSR), to avoid confusion with spatial or temporal downsampling factors.

If we were to increase the downsampling factor of the encoder by 2×, the latent grid size would then be 16×16, and we could increase the channel count by 4× to 32 channels, to maintain the same TSR. There are usually a few different configurations for a given TSR that perform roughly equally in terms of reconstruction quality, especially in the case of video, where we can separately control the temporal and spatial downsampling factors. If we change the TSR, however (by changing the downsampling factor without changing the channel count, or vice versa), this usually has a profound impact on both reconstruction quality and modelability.

From a purely mathematical perspective, this is surprising: if the latents are real-valued, the size of the grid and the number of channels shouldn’t matter, because the information capacity of a single number is already infinite (neatly demonstrated by Tupper’s self-referential formula). But of course, there are several practical limitations that restrict the amount of information a single component of the latent representation is able to carry:

• we use floating point representations of real numbers, which have finite precision;
• in many formulations, the encoder adds some amount of noise, which further limits effective precision;
• neural networks aren’t very good at learning highly nonlinear functions of their input.

The first one is obvious: if you represent a number with 32 bits (single precision), that is also the maximal number of bits of information it can possibly convey. Adding noise further reduces the number of usable bits, because some of the low-order digits will be overpowered by it.

The last limitation is actually the more restrictive one, but it is less well understood: isn’t the entire point of neural networks to learn nonlinear functions? That is true, but they are naturally biased towards learning relatively simple functions³⁷. This is usually a feature, not a bug, because it increases the probability of learning a function that generalises to unseen data. But if we are trying to compress a lot of information into a few numbers, that will likely require a high degree of nonlinearity. There are some ways to assist neural networks with learning more nonlinear functions (such as Fourier features³⁸), but in our setting, highly nonlinear mappings will actually negatively affect modelability: they obfuscate signal structure, so this is not a good solution. Representations with more components offer a better trade-off.

The same applies to discrete latent representations: the discretisation imposes a hard limit on the information content of the representation, but whether that capacity can be used efficiently depends chiefly on how expressive the encoder is, and how well the quantisation strategy works in practice (i.e. whether it achieves a high level of codebook utilisation by using the different codes as evenly as possible). I believe the most commonly used approach today is still the original VQ bottleneck from VQ-VAE¹², but a recent improvement which provides better gradient estimates using a “rotation trick”³⁹ seems promising in terms of codebook utilisation and end-to-end performance. Some alternatives without explicitly learnt codebooks have also gained traction recently: finite scalar quantisation (FSQ)⁴⁰, lookup-free quantisation (LFQ)⁴¹ and binary spherical quantisation (BSQ)⁴².

To recap, choosing the right TSR is key: a larger latent representation will yield better reconstruction quality (higher rate, lower distortion), but may negatively affect modelability. With a larger representation, there are simply more bits of information to model, therefore requiring more capacity in the generative model. In practice, this trade-off is usually tuned empirically. This can be an expensive affair, because there aren’t really any reliable proxy metrics for modelability that are cheap to compute. Therefore, it requires repeatedly training large enough generative models to get meaningful results.

Hansen-Estruch et al.⁴³ recently shared an extensive exploration of latent space capacity and the various factors that influence it (their key findings are clearly highlighted within the text). There is currently a trend toward increasing the spatial downsampling factor, and maintaining the TSR by also increasing the number of channels accordingly, in order to facilitate image and video generation at higher resolutions (e.g. 32× in LTX-Video⁴⁴ and GAIA-2⁴⁵, up to 64× in DCAE⁴⁶).

Curating and shaping the latent space

So far, we have talked about the capacity of latent representations, i.e. how many bits of information should go in them. It is just as important to control precisely which bits from the original input signals should be preserved in the latents, and how this information is presented. I will refer to the former as curating the latent space, and to the latter as shaping the latent space – the distinction is subtle, but important. Many regularisation strategies have been devised to shape, curate and control the capacity of latents. I will focus on the continuous case, but many of the same considerations apply to discrete latents as well.

VQGAN and KL-regularised latents

Rombach et al.²⁶ suggested two regularisation strategies for continuous latent spaces:

• Follow the original VQGAN recipe, reinterpreting the quantisation step as part of the decoder, rather than the encoder, to get a continuous representation (VQ-reg);
• Remove the quantisation step from the VQGAN recipe altogether, and replace it with a KL penalty, as in regular VAEs (KL-reg).

The idea of making only minimal changes to VQGAN to produce continuous latents (for use with diffusion models) is clever: the setup worked well for autoregressive models, and the quantisation during training serves as a safeguard to ensure that the latents don’t end up encoding too much information. However, as we discussed previously, this probably isn’t really necessary in most cases, because encoder expressivity is usually the limiting factor.

KL regularisation, on the other hand, is a core part of the traditional VAE setup: it is one of the two terms constituting the evidence lower bound (ELBO), which bounds the likelihood from below and enables VAE training to tractably (but indirectly) maximise the likelihood of the data. It encourages the latents to follow the imposed prior distribution (usually Gaussian). Crucially however, the ELBO is only truly a lower bound on the likelihood if there is no scaling hyperparameter in front of this term. Yet almost invariably, the KL term used to regularise continuous latent spaces is scaled down significantly (usually by several orders of magnitude), all but severing the connection with the variational inference context in which it originally arose.

The reason is simple: an unscaled KL term has too strong an effect, imposing a stringent limit on latent capacity and thus severely degrading reconstruction quality. The pragmatic response to that is naturally to scale down its relative contribution to the training loss. (Aside: in settings where one cares less about reconstruction quality, and more about semantic interpretability and disentanglement of the learnt representation, increasing the scale of the KL term can also be fruitful, as in beta-VAE³⁴.)

We are veering firmly into opinion territory here, but I feel there is currently quite a bit of magical thinking around the effect of the KL term. It is often suggested that this term encourages the latents to follow a Gaussian distribution, but with the scale factors that are typically used, this effect is way too weak to be meaningful. Even for “proper” VAEs, the aggregate posterior is rarely actually Gaussian^{47 48}.

All of this renders the “V” in “VAE” basically meaningless, in my opinion – its relevance is largely historical. We may as well drop it and talk about KL-regularised autoencoders instead, which more accurately reflects modern practice. The most important effect of the KL term in this setting is to supress outliers and constrain the scale of the latents to some extent. In other words: while the KL term is often presented as constraining capacity, the way it is used in practice mainly constrains the shape of the latents (but even that effect is relatively modest).

Tweaking reconstruction losses

The usual trio of reconstruction losses (regression, perceptual and adversarial) clearly plays an important role in maximising the quality of decoded signals, but it is also worth studying how these losses impact the latents, specifically in terms of curation (i.e. which information they learn to encode). As discussed in section 3, a good latent space in the visual domain makes abstraction of texture to some degree. How do these losses help achieve that?

A useful thought experiment is to consider what happens when we drop the perceptual and adversarial losses, retaining only the regression loss, as in traditional VAEs. This will tend to result in blurry reconstructions. Regression losses do not favour any particular kind of signal content by design, so in the case of images, they will focus on low-frequency content, simply because there is more of it. In natural images, the power of different spatial frequencies tends to be proportional to their inverse square – the higher the frequency, the less power (see my previous blog post for an illustrated analysis of this phenomenon). Since high frequencies constitute only a tiny fraction of the total signal power, the regression loss more strongly rewards accurate prediction of low frequencies than high ones. The relative perceptual importance of these high frequencies is much larger than the fraction of total signal power they represent, and blurry looking reconstructions are the well-known result.

Since texture is primarily composed of precisely these high frequencies which the regression loss largely ignores, we end up with a latent space that doesn’t make abstraction of texture, but rather erases textural information altogether. From a perceptual standpoint, that is a particularly undesirable form of latent space curation. This demonstrates the importance of the other two reconstruction loss terms, which ensure that the latents can encode some textural information.

If the regression loss has these undesirable properties, which require other loss terms to mitigate, perhaps we could drop it altogether? It turns out that’s not a great idea either, because the perceptual and adversarial losses are much harder to optimise and tend to have pathological local minima (they are usually based on pre-trained neural networks, after all). The regression loss acts as a sort of regulariser, continually providing guardrails against ending up in the bad parts of parameter space as training progresses.

Many strategies using different flavours of reconstruction losses have been suggested, and I won’t cover this space exhaustively, but here are a few examples from the literature, to give you an idea of the variety:

• The aforementioned DCAE⁴⁶ does not deviate too far from the original VQGAN recipe, only replacing the L2 regression loss (MSE) with L1 (MAE). It keeps the LPIPS perceptual loss and the PatchGAN⁴⁹ discriminator. It does however use multiple stages of training, with the adversarial loss only enabled in the last stage.
• ViT-VQGAN⁵⁰ combines two regression losses, the L2 loss and the logit-Laplace loss⁵¹, and uses the StyleGAN⁵² discriminator as well as the LPIPS perceptual loss.
• LTX-Video⁴⁴ introduces a video-aware loss based on the discrete wavelet transform (DWT), and uses a modified strategy for the adversarial loss which they call reconstruction-GAN.

As with many classic dishes, every chef still has their own recipe!

Representation learning vs. reconstruction

The design choices we have discussed so far usually impact not just reconstruction quality, but also the kind of latent space that is learnt. The reconstruction losses in particular do double duty: they ensure high-quality decoder output, and play an important role in curating the latent space as well. This raises the question whether it is actually desirable to kill two birds with one stone, as we have been doing. I would argue that the answer is no.

Learning a good compact representation for generative modelling on the one hand, and learning to decode that representation back to the input space on the other hand, are two separate tasks. Modern autoencoders are expected to learn to do both at once. The fact that this works reasonably well in practice is certainly a welcome convenience: training an autoencoder is already stage one of a two-stage training process, so ideally we wouldn’t want to complicate it any further (although having multiple autoencoder training stages is not unheard of^{46 53}). But this setup also needlessly conflates the two tasks, and some choices that are optimal for one task might not be for the other.

My colleagues and I actually made this point in a latent generative modelling paper⁵⁴, all the way back in 2019 (I recently tweeted about it as well). When the decoder is autoregressive, conflating the two tasks is particularly problematic, so we suggested using a separate non-autoregressive auxiliary decoder to provide the learning signal for the encoder. The main decoder is prevented from influencing the latent representation at all, by not backpropagating its gradients through to the encoder. It is therefore fully focused on maximising reconstruction quality, while the auxiliary decoder takes care of shaping and curating the latent space. All parts of the autoencoder can still be trained jointly, so the additional complexity is limited. The auxiliary decoder does of course increase the cost of training, but it can be discarded afterwards.

Although the idea of using autoregressive decoders in pixel space, as we did for that paper, has not aged well at all (to put it mildly), I believe using auxiliary decoders to separate the representation learning and reconstruction tasks is still a very relevant idea today. An auxiliary decoder that optimises a different loss, or that has a different architecture (or both), could provide a better learning signal for representation learning and result in better generative modelling performance.

Zhu et al.⁵⁵ recently came to the same conclusion (see section 2.1 of their paper), and constructed a discrete latent representation using K-means on DINOv2⁵⁶ features, combined with a separately trained decoder. Reusing representations learnt with self-supervised learning for generative modelling has historically been more common for audio models^{57 58 59 60} – perhaps because audio practitioners are already accustomed to the idea of training vocoders that turn predetermined intermediate representations (e.g. mel-spectrograms) back into audio waveforms.

Regularising for modelability

Shaping, curating and constraining the capacity of latents can all affect their modelability:

• Capacity constraints determine how much information is in the latents. The higher the capacity, the more powerful the generative model will have to be to adequately capture all of the information they contain.
• Shaping can be important to enable efficient modelling. The same information can be represented in many different ways, and some are easier to model than others. Scaling and standardisation are important to get right (especially for diffusion models), but higher-order statistics and correlation structure also matter.
• Curation influences modelability, because some kinds of information are much easier to model than others. If the latents encode information about unpredictable noise in the input signal, that will make them less predictable as well. Here’s an interesting tweet that demonstrates how this affects the Stable Diffusion XL VAE⁶¹:

The sdxl-VAE models a substantial amount of noise. Things we can't even see. It meticulously encodes the noise, uses precious bottleneck capacity to store it, then faithfully reconstructs it in the decoder.

I grabbed what I thought was a simple black vector circle on a white… pic.twitter.com/eK7ZtLJ6lc

— Rudy Gilman (@rgilman33) April 14, 2025

Here, I want to make the connection to the idea of \(\mathcal{V}\)-information, proposed by Xu et al.⁶², which extends the concept of mutual information to account for computational constraints (h/t @jxmnop on Twitter for bringing this work to my attention). In other words, the usability of information varies depending on how computationally challenging it is for an observer to discern, and we can try to quantify this. If a piece of information requires a powerful neural net to extract, the \(\mathcal{V}\)-information in the input is lower than in the case where a simple linear probe suffices – even when the absolute information content measured in bits is identical. Clearly, maximising the \(\mathcal{V}\)-information of the latent representation is desirable, in order to minimise the computational requirements for the generative model to be able to make sense of it. The rate-distortion-usefulness trade-off described by Tschannen et al., which I mentioned before³¹, supports the same conclusion.

As discussed earlier, the KL penalty probably doesn’t do quite as much to Gaussianise or otherwise smoothen the latent space as many seem to believe. So what can we do instead to make the latents easier to model?

• Use generative priors: co-train a (lightweight) latent generative model with the autoencoder, and make the latents easy to model by backpropagating the generative loss into the encoder, as in LARP⁶³ or CRT⁶⁴. This requires careful tuning of loss weights, because the generative loss and the reconstruction losses are at odds with each other: latents are easiest to model when they encode no information at all!

• Supervise with pre-trained representations: encourage the latents to be predictive of existing high-quality representations (e.g. DINOv2⁵⁶ features), as in VA-VAE⁶⁵, MAETok⁶⁶ or GigaTok⁶⁷.

• Encourage equivariance: make it so that certain transformations of the input (e.g. rescaling, rotations) produce corresponding latent representations that are transformed similarly, as in AuraEquiVAE, EQ-VAE³³ and AF-VAE⁶⁸. The figure from the EQ-VAE paper that I used in section 4 shows the profound impact that this constraint can have on the spatial smoothness of the latent space. Skorokhodov et al.⁶⁹ came to the same conclusion based on spectral analysis of latent spaces: equivariance regularisation makes the latent spectrum more similar to that of the pixel space inputs, which improves modelability.

This was just a small sample of possible regularisation strategies, all of which attempt to increase the \(\mathcal{V}\)-information of the latents in one way or another. This list is very far from exhaustive, both in terms of the strategies themselves and the works cited, so I encourage you to share any other relevant work you’ve come across in the comments!

Diffusion all the way down

A specific class of autoencoders for learning latent representations deserves a closer look: those with diffusion decoders. While a more typical decoder architecture features a feedforward network that directly outputs pixel values in one forward pass, and is trained adversarially, an alternative that’s gaining popularity is to use diffusion for the task of latent decoding (as well as for modelling the distribution of the latents). This impacts reconstruction quality, but it also affects what kind of representation is learnt.

SWYCC⁷⁰, \(\epsilon\)-VAE⁷¹ and DiTo⁷² are some recent works that explore this approach. They motivate this in a few different ways:

• latents learnt with diffusion decoders provide a more principled, theoretically grounded way of doing hierarchical generative modelling;
• they can be trained with just the MSE loss, which simplifies things and improves robustness (adversarial losses are pretty finicky to tune, after all);
• applying the principle of iterative refinement to decoding improves output quality.

I can’t really argue with any of these points, but I do want to point out one significant weakness of diffusion decoders: their computational cost, and the effect this has on decoder latency. I believe one of the key reasons that most commercially deployed diffusion models today are latent models, is that compact latent representations help us avoid iterative refinement in input space, which is slow and costly. Performing the iterative sampling procedure in latent space, and then going back to input space with a single forward pass at the end, is significantly faster.

With that in mind, reintroducing input-space iterative refinement for the decoding task looks to me like it largely defeats the point of the two-stage approach. If we are going to be paying that cost, we might as well opt for something like simple diffusion^{73 74} to scale up single-stage generative models instead.

Not so fast, you might say – can’t we use one of the many diffusion distillation methods to bring down the number of steps required? In a setting such as this one, with a very rich conditioning signal (i.e. the latent representation), these methods have indeed proven effective even down to the single-step sampling regime: the stronger the conditioning, the fewer steps are needed for high quality distillation results.

DALL-E 3’s consistency decoder⁷⁵ is a great practical example of this: they reused the Stable Diffusion²⁶ latent space and trained a new diffusion-based decoder, which was then distilled down to just two sampling steps using consistency distillation⁷⁶. While it is still more costly than the original adversarial decoder in terms of latency, the visual fidelity of the outputs is significantly improved.

Music2Latent⁷⁷ is another example of this approach, operating on spectrogram representations of music audio. Their autoencoder with a consistency decoder is trained end-to-end (unlike DALL-E 3’s, which reuses a pre-trained encoder), and is able to produce high-fidelity outputs in a single step. This means decoding once again requires only a single forward pass, as it does for adversarial decoders.

FlowMo⁵³ is an autoencoder with a diffusion decoder that uses a post-training stage to encourage mode-seeking behaviour. As mentioned before, for the task of decoding latent representations, dropping modes and focusing on realism over diversity is actually desirable, because it requires less model capacity and does not negatively impact perceptual quality. Adversarial losses tend to result in mode dropping, but diffusion-based losses do not. This two-stage training strategy enables the diffusion decoder to mimic this behaviour – although a significant number of sampling steps are still required, so the computational cost is considerably higher than for a typical adversarial decoder.

Some earlier works on diffusion autoencoders, such as Diff-AE⁷⁸ and DiffuseVAE⁷⁹, are more focused on learning high-level semantic representations in the vein of old-school VAEs, without topological structure, and with a focus on controllability and disentanglement. DisCo-Diff⁸⁰ sits somewhere in between, augmenting a diffusion model with a sequence of discrete latents, which can be modelled by an autoregressive prior.

Removing the need for adversarial training would certainly simplify things, so diffusion autoencoders are an interesting (and recently, quite popular) field of study in that regard. Still, it seems challenging to compete with adversarial decoders when it comes to latency, so I don’t think we are quite ready to abandon them. I very much look forward to an updated recipe that doesn’t require adversarial training, yet matches the current crop of adversarial decoders in terms of both visual quality and latency!

The tyranny of the grid

Digital representations of perceptual modalities are usually grid-structured, because they arise as uniformly sampled (and quantised) versions of the underlying physical signals. Images give rise to 2D grids of pixels, videos to 3D grids, and audio signals to 1D grids (i.e. sequences). The uniform sampling implies that there is a fixed quantum (i.e. distance or amount of time) between adjacent grid positions.

Perceptual signals also tend to be approximately stationary in time and space in a statistical sense. Combined with uniform sampling, this results in a rich topological structure, which we gratefully take advantage of when designing neural network architectures to process them: we use extensive weight sharing to benefit from invariance and equivariance properties, implemented through convolutions, recurrence and attention mechanisms.

Without a doubt, our ability to exploit this structure is one of the key reasons why we have been able to build machine learning models that are as powerful as they are. A corollary of this is that preserving this structure when designing latent spaces is a great idea. Our most powerful neural network designs architecturally depend on it, because they were originally built to process these digital signals directly. They will be better at processing latent representations instead, if those representations have the same kind of structure.

It also offers significant benefits for the autoencoders which learn to produce the latents: because of the stationarity, and because they only need to learn about local signal structure, they can be trained on smaller crops or segments of input signals. If we impose the right architectural constraints (limiting the receptive field of each position in the encoder and the decoder), they will generalise out of the box to larger grids than they were trained on. This has the potential to greatly reduce the training cost for the first stage.

It’s not all sunshine and rainbows though: we have discussed how perceptual signals are highly redundant, and unfortunately, this redundancy is unevenly distributed. Some parts of the signal might contain lots of detail that is perceptually salient, where others are almost devoid of information. In the image of a dog in a field that we used previously, consider a 100×100 pixel patch centered on the dog’s head, and then compare that to a 100×100 pixel patch in the top right corner of the image, which contains only the blue sky.

If we construct a latent representation which inherits the 2D grid structure of the input, and use it to encode this image, we will necessarily use the exact same amount of capacity to encode both of these patches. If we make the representation rich enough to capture all the relevant perceptual detail for the dog’s head, we will waste a lot of capacity to encode a similar-sized patch of sky. In other words, preserving the grid structure comes at a significant cost to the efficiency of the latent representation.

This is what I call the tyranny of the grid: our ability to process grid-structured data with neural networks is highly developed, and deviating from this structure adds complexity and makes the modelling task considerably harder and less hardware-friendly, so we generally don’t do this. But in terms of encoding efficiency, this is actually quite wasteful, because of the non-uniform distribution of perceptually salient information in audiovisual signals.

The Transformer architecture is actually relatively well-positioned to bolster a rebellion against this tyranny: while we often think of it as a sequence model, it is actually designed to process set-valued data, and any additional topological structure that relates elements of a set to each other is expressed through positional encoding. This makes deviating from a regular grid structure more practical than it is for convolutional or recurrent architectures. (Several years ago, my colleagues and I explored this idea for speech generation using variable-rate discrete representations⁸¹.)

Relaxing the topology of the latent space in the context of two-stage generative modelling appears to be gaining some traction lately:

• TiTok⁸² and FlowMo⁵³ learn sequence-structured latents from images, reducing the grid dimensionality from 2D to 1D. The development of large language models has given us extremely powerful sequence models, so this is a reasonable kind of structure to aim for.
• One-D-Piece⁸³, FlexTok⁸⁴ and Semanticist⁸⁵ do the same, but use a nested dropout mechanism⁸⁶ to induce a coarse-to-fine structure in the latent sequence. This in turn enables the sequence length to be adapted to the complexity of each individual input image, and to the level of detail required in the reconstruction. A few other mechanisms for adaptive 1D tokenisation have been proposed, e.g. ElasticTok⁸⁷ and ALIT⁸⁸. CAT⁸⁹ also explores this kind of adaptivity, but still maintains a 2D grid structure and only adapts its resolution.
• TokenSet⁹⁰ goes a step further and uses an autoencoder that produces “bags of tokens”, abandoning the grid completely.

Aside from CAT, all of these have in common that they learn latent spaces that are considerably more semantically high-level than the ones we have mostly been talking about so far. In terms of abstraction level, they probably sit somewhere in the middle between “advanced pixels” and the vector-valued latents of old-school VAEs. In fact, FlexTok and Semanticist’s 1D sequence encoders expect low-level latents from an existing 2D grid-structured encoder as input, literally building an additional abstraction level on top of a pre-existing low-level latent space. TiTok and One-D-Piece also make use of an existing 2D grid-structured latent space as part of a multi-stage training approach. A related idea is to reuse the language domain as a high-level latent representation for images⁹¹.

In the discrete setting, some work has investigated whether commonly occurring patterns of tokens in a grid can be combined into larger sub-units, using ideas from language tokenisation: DiscreTalk⁹² is an early example in the speech domain, using SentencePiece⁹³ on top of VQ tokens. Zhang et al.’s BPE Image Tokenizer⁹⁴ is a more recent incarnation of this idea, using an enhanced byte-pair encoding⁹⁵ algorithm on VQGAN tokens.

Latents for other modalities

We have chiefly focused on the visual domain so far, only briefly mentioning audio in some places. This is because learning latents for images is something we have gotten pretty good at, and image generation with the two-stage approach has been extensively studied (and productionised!) in recent years. We have a well-developed body of research on perceptual losses, and a host of discriminator architectures that enable adversarial training to focus on perceptually relevant image content.

With video, we remain in the visual domain, but a temporal dimension is introduced, which presents some challenges. One could simply reuse image latents and extract them frame-by-frame to get a latent video representation, but this would likely lead to temporal artifacts (like flickering). More importantly, it does not allow for taking advantage of temporal redundancy. I believe our tools for spatiotemporal latent representation learning are considerably less well developed, and how to account for the human perception of motion to improve efficiency is less well understood for now. This is in spite of the fact that video compression algorithms all make use of motion estimation to improve efficiency.

The same goes for audio: while there has been significant adoption of the two-stage recipe^{96 97 98}, there seems to be considerably less of a broad consensus on the necessary modifications to make it work for this modality. As mentioned before, for audio it is also much more common to reuse representations learnt with self-supervised learning.

What about language? This is not a perceptual modality, but maybe the two-stage approach could improve the efficiency of large language models as well? This is not straightforward, as it turns out. Language is inherently much less compressible than perceptual signals: it developed as a means of efficient communication, so there is considerably less redundancy. Which is not to say there isn’t any: Shannon famously estimated English to be 50% redundant⁹⁹. But recall that images, audio and video can be compressed by several orders of magnitude with relatively minimal perceptual distortion, which is not possible with language without losing nuance or important semantic information.

Tokenisers used for language models tend to be lossless (e.g. BPE⁹⁵, SentencePiece⁹³), so the resulting tokens aren’t usually viewed as “latents” (Byte Latent Transformer¹⁰⁰ does use this framing for its dynamic tokenisation strategy, however). The relative lack of redundancy in language has not stopped people from trying to learn lossy higher-level representations, though! The techniques used for perceptual signals may not carry over, but several other methods to learn representations at the sentence or paragraph level have been explored^{101 102 103}.

Will end-to-end win in the end?

When deep learning rose to prominence, the dominant narrative was that we would replace handcrafted features with end-to-end learning wherever possible. Jointly learning all processing stages would allow these stages to co-adapt and cooperate to maximise performance, while also simplifying things from an engineering perspective. This is more or less what ended up happening in computer vision and speech processing. In that light, it is rather ironic that the dominant generative modelling paradigm for perceptual signals today is a two-stage approach. Both stages tend to be learnt, but it’s not exactly end-to-end!

Text-to-image, text-to-video and text-to-audio models deployed in products today are, for the most part, using intermediate latent representations. It is worth pondering whether this is a temporary status quo, or likely to continue to be the case. Two-stage training does introduce a significant amount of complexity after all, and apart from being more elegant, end-to-end learning would help ensure that all parts of a system are perfectly in tune with a single, overarching objective.

As discussed, iterative refinement in the input space is slow and costly, and I think this is likely to remain the case for a while longer – especially as we continue to ramp up the quality, resolution and/or length of the generated signals. The benefits of latents for training efficiency and sampling latency are not something we are likely to be happy to give up on, and there are currently no viable alternatives that have been demonstrated to work at scale. This is somewhat of a contentious point, because some researchers seem to believe the time has come to move towards an end-to-end approach. Personally, I think it’s too early.

So, when will we be ready to move back to single-stage generative models? Methods like simple diffusion^{73 74}, Ambient Space Flow Transformers¹⁰⁴ and PixelFlow¹⁰⁵ have shown that this already works pretty well, even at relatively high resolutions – it just isn’t very cost-effective yet. But hardware keeps getting better and faster at an incredible rate, so I suspect we will eventually reach a point where the relative inefficiency of input-space models starts to be economically preferable over the increased engineering complexity of latent-space models. When exactly this will happen depends on the modality, the rate of hardware improvements and research progress, so I will stop short of making a concrete prediction.

It used to be the case that we needed latents to ensure that generative models would focus on learning about perceptually relevant signal content, while ignoring entropy that is not visually salient. Recall that the likelihood loss in input space is particularly bad at this, and switching to measuring likelihood in latent space dramatically improved results obtained with likelihood-based models¹². Arguably, this is no longer the case, because we have figured out how to perceptually re-weight the likelihood loss function for both autoregressive and diffusion models^{106 107}, removing an important obstacle to scaling. But in spite of that, the computational efficiency benefits of latent-space models remain as relevant as ever.

A third alternative, which I’ve only briefly mentioned so far, is the resolution cascade approach. This requires no representation learning, but still splits up the generative modelling problem into multiple stages. Some early commercial models used this approach, but it seems to have fallen out of favour. I believe this is because the division of labour between the different stages is suboptimal – upsampling models have to do too much of the work, and this makes them more prone to error accumulation across stages.

Closing thoughts

To wrap up, here are some key takeaways:

• Latents for generative modelling are usually quite unlike VAE latents from way back when: it makes more sense to think of them as advanced pixels than as high-level semantic representations.
• Having two stages enables us to have different loss functions for each, and significantly improves computational efficiency by avoiding iterative sampling in the input space.
• Latents add complexity, but the computational efficiency benefits are large enough for us to tolerate this complexity – at least for now.
• Three main aspects to consider when designing latent spaces are capacity (how many bits of information are encoded in the latents), curation (which bits from the input signals are retained) and shape (how this information is presented).
• Preserving structure (i.e. topology, statistics) in the latent space is important to make it easy to model, even if this is sometimes worse from an efficiency perspective.
• The V in VAE is vestigial: the autoencoders used in two-stage latent generative models are really KL-regularised AEs, and usually barely regularised at that.
• The combination of a regression loss, a perceptual loss and an adversarial loss is surprisingly entrenched. Almost all modern implementations of autoencoders for latent representation learning are variations on this theme.
• Representation learning and reconstruction are two separate tasks, and while it is convenient to do both at once, this might not be optimal.

Several years ago, I wrote on this blog that representation learning has an important role to play in generative modelling. Though it wasn’t exactly prophetic at the time (VQ-VAE had already existed for three years), the focus in the representation learning community back then was firmly on high-level semantic representations that are useful for discriminative tasks. It was a lot more common to hear about generative modelling in service of representation learning, than vice versa. How the tables have turned! The arrival of VQGAN¹⁶ a few months later was probably what cemented the path we now find ourselves on, and made two-stage generative modelling go mainstream.

Thank you for reading what ended up being my longest blog post yet! I’m keen to hear your thoughts here in the comments, on Twitter (yes, that’s what I’m calling it), or on BlueSky.

If you would like to cite this post in an academic context, you can use this BibTeX snippet:

@misc{dieleman2025latents,
  author = {Dieleman, Sander},
  title = {Generative modelling in latent space},
  url = {https://sander.ai/2025/04/15/latents.html},
  year = {2025}
}

Acknowledgements

Thanks to my colleagues at Google DeepMind for various discussions, which continue to shape my thoughts on this topic! Thanks also to Stefan Baumann, Rami Seid, Charles Foster, Ethan Smith, Simo Ryu, Theodoros Kouzelis and Jack Gallagher.

References

A bit of signal processing swiftly reveals that diffusion models and autoregressive models aren’t all that different: diffusion models of images perform approximate autoregression in the frequency domain!

This blog post is also available as a Python notebook in Google Colab , with the code used to produce all the plots and animations.

Last year, I wrote a blog post describing various different perspectives on diffusion. The idea was to highlight a number of connections between diffusion models and other classes of models and concepts. In recent months, I have given a few talks where I discussed some of these perspectives. My talk at the EEML 2024 summer school in Novi Sad, Serbia, was recorded and is available on YouTube. Based on the response I got from this talk, the link between diffusion models and autoregressive models seems to be particularly thought-provoking. That’s why I figured it could be useful to explore this a bit further.

In this blog post, I will unpack the above claim, and try to make it obvious that this is the case, at least for visual data. To make things more tangible, I decided to write this entire blog post in the form of a Python notebook (using Google Colab). That way, you can easily reproduce the plots and analyses yourself, and modify them to observe what happens. I hope this format will also help drive home the point that this connection between diffusion models and autoregressive models is “real”, and not just a theoretical idealisation that doesn’t hold up in practice.

In what follows, I will assume a basic understanding of diffusion models and the core concepts behind them. If you’ve watched the talk I linked above, you should be able to follow along. Alternatively, the perspectives on diffusion blog post should also suffice as preparatory reading. Some knowledge of the Fourier transform will also be helpful.

Below is an overview of the different sections of this post. Click to jump directly to a particular section.

1. Two forms of iterative refinement
2. A spectral view of diffusion
3. What about sound?
4. Unstable equilibrium
5. Closing thoughts
6. Acknowledgements
7. References

Two forms of iterative refinement

Autoregression and diffusion are currently the two dominant generative modelling paradigms. There are many more ways to build generative models: flow-based models and adversarial models are just two possible alternatives (I discussed a few more in an earlier blog post).

Both autoregression and diffusion differ from most of these alternatives, by splitting up the difficult task of generating data from complex distributions into smaller subtasks that are easier to learn. Autoregression does this by casting the data to be modelled into the shape of a sequence, and recursively predicting one sequence element at a time. Diffusion instead works by defining a corruption process that gradually destroys all structure in the data, and training a model to learn to invert this process step by step.

This iterative refinement approach to generative modelling is very powerful, because it allows us to construct very deep computational graphs for generation, without having to backpropagate through them during training. Indeed, both autoregressive models and diffusion models learn to perform a single step of refinement at a time – the generative process is not trained end-to-end. It is only when we try to sample from the model that we connect all these steps together, by sequentially performing the subtasks: predicting one sequence element after another in the case of autoregression, or gradually denoising the input step-by-step in the case of diffusion.

Because this underlying iterative approach is common to both paradigms, people have often sought to connect the two. One could frame autoregression as a special case of discrete diffusion, for example, with a corruption process that gradually replaces tokens by “mask tokens” from right to left, eventually ending up with a fully masked sequence. In the next few sections, we will do the opposite, framing diffusion as a special case of autoregression, albeit approximate.

Today, most language models are autoregressive, while most models of images and video are diffusion-based. In many other application domains (e.g. protein design, planning in reinforcement learning, …), diffusion models are also becoming more prevalent. I think this dichotomy, which can be summarised as “autoregression for language, and diffusion for everything else”, is quite interesting. I have written about it before, and I will have more to say about it in a later section of this post.

A spectral view of diffusion

Image spectra

When diffusion models rose to prominence for image generation, people noticed quite quickly that they tend to produce images in a coarse-to-fine manner. The large-scale structure present in the image seems to be decided in earlier denoising steps, whereas later denoising steps add more and more fine-grained details.

To formalise this observation, we can use signal processing, and more specifically spectral analysis. By decomposing an image into its constituent spatial frequency components, we can more precisely tease apart its coarse- and fine-grained structure, which correspond to low and high frequencies respectively.

We can use the 2D Fourier transform to obtain a frequency representation of an image. This representation is invertible, i.e. it contains the same information as the pixel representation – it is just organised in a different way. Like the pixel representation, it is a 2D grid-structured object, with the same width and height as the original image, but the axes now correspond to horizontal and vertical spatial frequencies, rather than spatial positions.

To see what this looks like, let’s take some images and visualise their spectra.

Shown above on the first row are four images from the Imagenette dataset, a subset of the ImageNet dataset (I picked it because it is relatively fast to load).

The Fourier transform is typically complex-valued, so the next two rows visualise the magnitude and the phase of the spectrum respectively. Because the magnitude varies greatly across different frequencies, its logarithm is shown. The phase is an angle, which varies between \(-\pi\) and \(\pi\). Note that we only calculate the spectrum for the green colour channel – we could calculate it for the other two channels as well, but they would look very similar.

The centre of the spectrum corresponds to the lowest spatial frequencies, and the frequencies increase as we move outward to the edges. This allows us to see where most of the energy in the input signal is concentrated. Note that by default, it is the other way around (low frequencies in the corner, high frequencies in the middle), but np.fft.fftshift allows us to swap these, which yields a much nicer looking visualisation that makes the structure of the spectrum more apparent.

A lot of interesting things can be said about the phase structure of natural images, but in what follows, we will primarily focus on the magnitude spectrum. The square of the magnitude is the power, so in practice we often look at the power spectrum instead. Note that the logarithm of the power spectrum is simply that of the magnitude spectrum, multiplied by two.

Looking at the spectra, we now have a more formal way to reason about different feature scales in images, but that still doesn’t explain why diffusion models exhibit this coarse-to-fine behaviour. To see why this happens, we need to examine what a typical image spectrum looks like. To do this, we will make abstraction of the directional nature of frequencies in 2D space, simply by slicing the spectrum along a certain angle, rotating that slice all around, and then averaging the slices across all rotations. This yields a one-dimensional curve: the radially averaged power spectral density, or RAPSD.

Below is an animation that shows individual directional slices of the 2D spectrum on a log-log plot, which are averaged to obtain the RAPSD.

Let’s see what that looks like for the four images above. We will use the pysteps library, which comes with a handy function to calculate the RAPSD in one go.

The RAPSD is best visualised on a log-log plot, to account for the large variation in scale. We chop off the so-called DC component (with frequency 0) to avoid taking the logarithm of 0.

Another thing this visualisation makes apparent is that the curves are remarkably close to being straight lines. A straight line on a log-log plot implies that there might be a power law lurking behind all of this.

Indeed, this turns out to be the case: natural image spectra tend to approximately follow a power law, which means that the power \(P(f)\) of a particular frequency \(f\) is proportional to \(f^{-\alpha}\), where \(\alpha\) is a parameter^{1 2 3}. In practice, \(\alpha\) is often remarkably close to 2 (which corresponds to the spectrum of pink noise in two dimensions).

We can get closer to the “typical” RAPSD by taking the average across a bunch of images (in the log-domain).

As I’m sure you will agree, that is pretty unequivocally a power law!

To estimate the exponent \(\alpha\), we can simply use linear regression in log-log space. Before proceeding however, it is useful to resample our averaged RAPSD so the sample points are linearly spaced in log-log space – otherwise our fit will be dominated by the high frequencies, where we have many more sample points.

We obtain an estimate \(\hat{\alpha} = 2.454\), which is a bit higher than the typical value of 2. As far as I understand, this can be explained by the presence of man-made objects in many of the images we used, because they tend to have smooth surfaces and straight angles, which results in comparatively more low-frequency content and less high-frequency content compared to images of nature. Let’s see what our fit looks like.

Noisy image spectra

A crucial aspect of diffusion models is the corruption process, which involves adding Gaussian noise. Let’s see what this does to the spectrum. The first question to ask is: what does the spectrum of noise look like? We can repeat the previous procedure, but replace the image input with standard Gaussian noise. For contrast, we will visualise the spectrum of the noise alongside that of the images from before.

The RAPSD of Gaussian noise is also a straight line on a log-log plot; but a horizontal one, rather than one that slopes down. This reflects the fact that Gaussian noise contains all frequencies in equal measure. The Fourier transform of Gaussian noise is itself Gaussian noise, so its power must be equal across all frequencies in expectation.

When we add noise to the images and look at the spectrum of the resulting noisy images, we see a hinge shape:

Why does this happen? Recall that the Fourier transform is linear: the Fourier transform of the sum of two things, is the sum of the Fourier transforms of those things. Because the power of the different frequencies varies across orders of magnitude, one of the terms in this sum tends to drown out the other. This is what happens at low frequencies, where the image spectrum dominates, and hence the green curve overlaps with the red curve. At high frequencies on the other hand, the noise spectrum dominates, and the green curve overlaps with the blue curve. In between, there is a transition zone where the power of both spectra is roughly matched.

If we increase the variance of the noise by scaling the noise term, we increase its power, and as a result, its RAPSD will shift upward (which is also a consequence of the linearity of the Fourier transform). This means a smaller part of the image spectrum now juts out above the waterline: the increasing power of the noise looks like the rising tide!

At this point, I’d like to revisit a diagram from the perspectives on diffusion blog post, where I originally drew the connection between diffusion and autoregression in frequency space, which is shown below.

These idealised plots of the spectra of images, noise, and their superposition match up pretty well with the real versions. When I originally drew this, I didn’t actually realise just how closely this reflects reality!

What these plots reveal is an approximate equivalence (in expectation) between adding noise to images, and low-pass filtering them. The noise will drown out some portion of the high frequencies, and leave the low frequencies untouched. The variance of the noise determines the cut-off frequency of the filter. Note that this is the case only because of the characteristic shape of natural image spectra.

The animation below shows how the spectrum changes as we gradually add more noise, until it eventually overpowers all frequency components, and all image content is gone.

Diffusion

With this in mind, it becomes apparent that the corruption process used in diffusion models is actually gradually filtering out more and more high-frequency information from the input image, and the different time steps of the process correspond to a frequency decomposition: basically an approximate version of the Fourier transform!

Since diffusion models themselves are tasked with reversing this corruption process step-by-step, they end up roughly predicting the next higher frequency component at each step of the generative process, given all preceding (lower) frequency components. This is a soft version of autoregression in frequency space, or if you want to make it sound fancier, approximate spectral autoregression.

To the best of my knowledge, Rissanen et al. (2022)⁴ were the first to apply this kind of analysis to diffusion in the context of generative modelling (see §2.2 in the paper). Their work directly inspired this blog post.

In many popular formulations of diffusion, the corruption process does not just involve adding noise, but also rescaling the input to keep the total variance within a reasonable range (or constant, in the case of variance-preserving diffusion). I have largely ignored this so far, because it doesn’t materially change anything about the intuitive interpretation. Scaling the input simply results in the RAPSD shifting up or down a bit.

Which frequencies are modelled at which noise levels?

There seems to be a monotonic relationship between noise levels and spatial frequencies (and hence feature scales). Can we characterise this quantitatively?

We can try, but it is important to emphasise that this relationship is only really valid in expectation, averaged across many images: for individual images, the spectrum will not be a perfectly straight line, and it will not typically be monotonically decreasing.

Even if we ignore all that, the “elbow” of the hinge-shaped spectrum of a noisy image is not very sharp, so it is clear that there is quite a large transition zone where we cannot unequivocally say that a particular frequency is dominated by either signal or noise. So this is, at best, a very smooth approximation to the “hard” autoregression used in e.g. large language models.

Keeping all of that in mind, let us construct a mapping from noise levels to frequencies for a particular diffusion process and a particular image distribution, by choosing a signal-to-noise ratio (SNR) threshold, below which we will consider the signal to be undetectable. This choice is quite arbitrary, and we will just have to choose a value and stick with it. We can choose 1 to keep things simple, which means that we consider the signal to be detectable if its power is equal to or greater than the power of the noise.

Consider a Gaussian diffusion process for which \(\mathbf{x}_t = \alpha(t)\mathbf{x}_0 + \sigma(t) \mathbf{\varepsilon}\), with \(\mathbf{x}_0\) an example from the data distribution, and \(\mathbf{\varepsilon}\) standard Gaussian noise.

Let us define \(\mathcal{R}[\mathbf{x}](f)\) as the RAPSD of an image \(\mathbf{x}\) evaluated at frequency \(f\). We will call the SNR threshold \(\tau\). If we consider a particular time step \(t\), then assuming the RAPSD is monotonically decreasing, we can define the maximal detectable frequency \(f_\max\) at this time step in the process as the maximal value of \(f\) for which:

Recall that the Fourier transform is a linear operator, and \(\mathcal{R}\) is a radial average of the square of its magnitude. Therefore, scaling the input to \(\mathcal{R}\) by a real value means the output gets scaled by its square. We can use this to simplify things:

We can further simplify this by noting that \(\forall f: \mathcal{R}[\mathbf{\varepsilon}](f) = 1\):

To construct such a mapping in practice, we first have to choose a diffusion process, which gives us the functional form of \(\sigma(t)\) and \(\alpha(t)\). To keep things simple, we can use the rectified flow⁵ / flow matching⁶ process, as used in Stable Diffusion 3⁷, for which \(\sigma(t) = t\) and \(\alpha(t) = 1 - t\). Combined with \(\tau = 1\), this yields:

With these choices, we can now determine the shape of \(f_\max(t)\) and visualise it.

The frequencies here are relative: if the bandwidth of the signal is 1, then 0.5 corresponds to the Nyquist frequency, i.e. the maximal frequency that is representable with the given bandwidth.

Note that all representable frequencies are detectable at time steps near 0. As \(t\) increases, so does the noise level, and hence \(f_\max\) starts dropping, until it eventually reaches 0 (no detectable signal frequencies are left) close to \(t = 1\).

What about sound?

All of the analysis above hinges on the fact that spectra of natural images typically follow a power law. Diffusion models have also been used to generate audio^{8 9}, which is the other main perceptual modality besides the visual. A very natural question to ask is whether the same interpretation makes sense in the audio domain as well.

To establish that, we will grab a dataset of typical audio recordings that we might want to build a generative model of: speech and music.

Along with each audio player, a spectrogram is shown: this is a time-frequency representation of the sound, which is obtained by applying the Fourier transform to short overlapping windows of the waveform and stacking the resulting magnitude vectors together in a 2D matrix.

For the purpose of comparing the spectrum of sound with that of images, we will use the 1-dimensional analogue of the RAPSD, which is simply the squared magnitude of the 1D Fourier transform.

These are a lot noisier than the image spectra, which is not surprising as these are not averaged over directions, like the RAPSD is. But aside from that, they don’t really look like straight lines either – the power law shape is nowhere to be seen!

I won’t speculate about why images exhibit this behaviour and sound seemingly doesn’t, but it is certainly interesting (feel free to speculate away in the comments!). To get a cleaner view, we can again average the spectra of many clips in the log domain, as we did with the RAPSDs of images.

Definitely not a power law. More importantly, it is not monotonic, so adding progressively more Gaussian noise to this does not obfuscate frequencies in descending order: the “diffusion is just spectral autoregression” meme does not apply to audio waveforms!

The average spectrum of our dataset exhibits a peak around 300-400 Hz. This is not too far off the typical spectrum of green noise, which has more energy in the region of 500 Hz. Green noise is supposed to sound like “the background noise of the world”.

As the animation above shows, the different frequencies present in audio signals still get filtered out gradually from least powerful to most powerful, because the spectrum of Gaussian noise is still flat, just like in the image domain. But as the audio spectrum does not monotonically decay with increasing frequency, the order is not monotonic in terms of the frequencies themselves.

What does this mean for diffusion in the waveform domain? That’s not entirely clear to me. It certainly makes the link with autoregressive models weaker, but I’m not sure if there are any negative implications for generative modelling performance.

One observation that does perhaps indicate that this is the case, is that a lot of diffusion models of audio described in the literature do not operate directly in the waveform domain. It is quite common to first extract some form of spectrogram (as we did earlier), and perform diffusion in that space, essentially treating it like an image^{10 11 12}. Note that spectrograms are a somewhat lossy representation of sound, because phase information is typically discarded.

To understand the implications of this for diffusion models, we will extract log-scaled mel-spectrograms from the sound clips we have used before. The mel scale is a nonlinear frequency scale which is intended to be perceptually uniform, and which is very commonly used in spectral analysis of sound.

Next, we will interpret these spectrograms as images and look at their spectra. Taking the spectrum of a spectrum might seem odd – some of you might even suggest that it is pointless, because the Fourier transform is its own inverse! But note that there are a few nonlinear operations happening in between: taking the magnitude (discarding the phase information), mel-binning and log-scaling. As a result, this second Fourier transform doesn’t just undo the first one.

It seems like the power law has resurfaced! We can look at the average in the log-domain again to get a smoother curve.

I found this pretty surprising. I actually used to object quite strongly to the idea of treating spectrograms as images, as in this tweet in response to Riffusion, a variant of Stable Diffusion finetuned on spectrograms:

Me: "NOOO, you can't just treat spectrograms as images, the frequency and time axes have completely different semantics, there is no locality in frequency and ..."

These guys: "Stable diffusion go brrr" https://t.co/Akv8aZl8Rv

— Sander Dieleman (@sedielem) December 15, 2022

… but I have always had to concede that it seems to work pretty well in practice, and perhaps the fact that spectrograms exhibit power-law spectra is one reason why.

There is also an interesting link with mel-frequency cepstral coefficients (MFCCs), a popular feature representation for speech and music processing which predates the advent of deep learning. These features are constructed by taking the discrete cosine transform (DCT) of a mel-spectrogram. The resulting spectrum-of-a-spectrum is often referred to as the cepstrum.

So with this approach, perhaps the meme applies to sound after all, albeit with a slight adjustment: diffusion on spectrograms is just cepstral autoregression.

Unstable equilibrium

So far, we have talked about a spectral perspective on diffusion, but we have not really discussed how it can be used to explain why diffusion works so well for images. The fact that this interpretation is possible for images, but not for some other domains, does not automatically imply that the method should also work better.

However, it does mean that the diffusion loss, which is a weighted average across all noise levels, is also implicitly a weighted average over all spatial frequencies in the image domain. Being able to individually weight these frequencies in the loss according to their relative importance is key, because the sensitivity of the human visual system to particular frequencies varies greatly. This effectively makes the diffusion training objective a kind of perceptual loss, and I believe it largely explains the success of diffusion models in the visual domain (together with classifier-free guidance).

Going beyond images, one could use the same line of reasoning to try and understand why diffusion models haven’t really caught on in the domain of language modelling so far (I wrote more about this last year). The interpretation in terms of a frequency decomposition is not really applicable there, and hence being able to change the relative weighting of noise levels in the loss doesn’t quite have the same impact on the quality of generated outputs.

For language modelling, autoregression is currently the dominant modelling paradigm, and while diffusion-based approaches have been making inroads recently^{13 14 15}, a full-on takeover does not look like it is in the cards in the short term.

This results in the following status quo: we use autoregression for language, and we use diffusion for pretty much everything else. Of course, I realise that I have just been arguing that these two approaches are not all that different in spirit. But in practice, their implementations can look quite different, and a lot of knowledge and experience that practitioners have built up is specific to each paradigm.

To me, this feels like an unstable equilibrium, because the future is multimodal. We will ultimately want models that natively understand language, images, sound and other modalities mixed together. Grafting these two different modelling paradigms together to construct multimodal models is effective to some extent, and certainly interesting from a research perspective, but it brings with it an increased level of complexity (i.e. having to master two different modelling paradigms) which I don’t believe practitioners will tolerate in the long run.

So in the longer term, it seems plausible that we could go back to using autoregression across all modalities, perhaps borrowing some ideas from diffusion in the process^{16 17}. Alternatively, we might figure out how to build multimodal diffusion models for all modalities, including language. I don’t know which it is going to be, but both of those outcomes ultimately seem more likely than the current situation persisting.

One might ask, if diffusion is really just approximate autoregression in frequency space, why not just do exact autoregression in frequency space instead, and maybe that will work just as well? That would mean we can use autoregression across all modalities, and resolve the “instability” in one go. Nash et al. (2021)¹⁸, Tian et al. (2024)¹⁶ and Mattar et al. (2024)¹⁹ explore this direction.

There is a good reason not to take this shortcut, however: the diffusion sampling procedure is exceptionally flexible, in ways that autoregressive sampling is not. For example, the number of sampling steps can be chosen at test time (this isn’t impossible for autoregressive models, but it is much less straightforward to achieve). This flexibility also enables various distillation methods to reduce the number of steps required, and classifier-free guidance to improve sample quality. Before we do anything rash and ditch diffusion altogether, we will probably want to figure out a way to avoid having to give up some of these benefits.

Closing thoughts

When I first had a closer look at the spectra of real images myself, I realised that the link between diffusion models and autoregressive models is even stronger than I had originally thought – in the image domain, at least. This is ultimately why I decided to write this blog post in a notebook, to make it easier for others to see this for themselves as well. More broadly speaking, I find that learning by “doing” has a much more lasting effect than learning by reading, and hopefully making this post interactive can help with that.

There are of course many other ways to connect the two modelling paradigms of diffusion and autoregression, which I won’t go into here, but it is becoming a rather popular topic of inquiry^{20 21 22}.

If you enjoyed this post, I strongly recommend also reading Rissanen et al. (2022)’s paper on generative modelling with inverse heat dissipation⁴, which inspired it.

This blog-post-in-a-notebook was an experiment, so any feedback on the format is very welcome! It’s a bit more work, but hopefully some readers will derive some benefit from it. If there are enough of you, perhaps I will do more of these in the future. Please share your thoughts in the comments!

To wrap up, below are some low-effort memes I made when I should have been working on this blog post instead.

The interpretation of diffusion as autoregression in the frequency domain seems to be stirring up a lot of thought! (I may or may not have a new blog post in the works 🧐) pic.twitter.com/XSxP27pKSt

— Sander Dieleman (@sedielem) August 4, 2024

It's so much easier to tweet low-effort memes which assert that diffusion is just autoregression in frequency space, than it is to write a blog post about it 🤷 (but I'm doing both!) pic.twitter.com/snLQavtZBf

— Sander Dieleman (@sedielem) August 22, 2024

If you would like to cite this post in an academic context, you can use this BibTeX snippet:

@misc{dieleman2024spectral,
  author = {Dieleman, Sander},
  title = {Diffusion is spectral autoregression},
  url = {https://sander.ai/2024/09/02/spectral-autoregression.html},
  year = {2024}
}

Acknowledgements

Thanks to my colleagues at Google DeepMind for various discussions, which continue to shape my thoughts on this topic! In particular, thanks to Robert Riachi, Ruben Villegas and Daniel Zoran.

References

The noise schedule is a key design parameter for diffusion models. It determines how the magnitude of the noise varies over the course of the diffusion process. In this post, I want to make the case that this concept sometimes confuses more than it elucidates, and we might be better off if we reframed things without reference to noise schedules altogether.

All of my blog posts are somewhat subjective, and I usually don’t shy away from highlighting my favourite ideas, formalisms and papers. That said, this one is probably a bit more opinionated still, maybe even a tad spicy! Probably the spiciest part is the title, but I promise I will explain my motivation for choosing it. At the same time, I also hope to provide some insight into the aspects of diffusion models that influence the relative importance of different noise levels, and why this matters.

This post will be most useful to readers familiar with the basics of diffusion models. If that’s not you, don’t worry; I have a whole series of blog posts with references to bring you up to speed! As a starting point, check out Diffusion models are autoencoders and Perspectives on diffusion. Over the past few years, I have written a few more on specific topics as well, such as guidance and distillation. A list of all my blog posts can be found here.

Below is an overview of the different sections of this post. Click to jump directly to a particular section.

1. Noise schedules: a whirlwind tour
2. Noise levels: focusing on what matters
3. Model design choices: what might tip the balance?
4. Noise schedules are a superfluous abstraction
5. Adaptive weighting mechanisms
6. Closing thoughts
7. Acknowledgements
8. References

Noise schedules: a whirlwind tour

Most descriptions of diffusion models consider a process that gradually corrupts examples of a data distribution with noise. The task of the model is then to learn how to undo the corruption. Additive Gaussian noise is most commonly used as the corruption method. This has the nice property that adding noise multiple times in sequence yields the same outcome (in a distributional sense) as adding noise once with a higher standard deviation. The total standard deviation is found as \(\sigma = \sqrt{ \sum_i \sigma_i^2}\), where \(\sigma_1, \sigma_2, \ldots\) are the standard deviations of the noise added at each point in the sequence.

Therefore, at each point in the corruption process, we can ask: what is the total amount of noise that has been added so far – what is its standard deviation? We can write this as \(\sigma(t)\), where \(t\) is a time variable that indicates how far the corruption process has progressed. This function \(\sigma(t)\) is what we typically refer to as the noise schedule. Another consequence of this property of Gaussian noise is that we can jump forward to any point in the corruption process in a single step, simply by adding noise with standard deviation \(\sigma(t)\) to a noiseless input example. The distribution of the result is exactly the same as if we had run the corruption process step by step.

In addition to adding noise, the original noiseless input is often rescaled by a time-dependent scale factor \(\alpha(t)\) to stop it from growing uncontrollably. Given an example \(\mathbf{x}_0\), we can turn it into a noisy example \(\mathbf{x}_t = \alpha(t) \mathbf{x}_0 + \sigma(t) \varepsilon\), where \(\varepsilon \sim \mathcal{N}(0, 1)\).

• The most popular formulation of diffusion models chooses \(\alpha(t) = \sqrt{1 - \sigma(t)^2}\), which also requires that \(\sigma(t) \leq 1\). This is because if we assume \(\mathrm{Var}[\mathbf{x}_0] = 1\), we can derive that \(\mathrm{Var}[\mathbf{x}_t] = 1\) for all \(t\). In other words, this choice is variance-preserving: the total variance (of the signal plus the added noise) is \(1\) at every step of the corruption process. In the literature, this is referred to as VP diffusion¹. While \(\mathrm{Var}[\mathbf{x}_0] = 1\) isn’t always true in practice (for example, image pixels scaled to \([-1, 1]\) will have a lower variance), it’s often close enough that things still work well in practice.

• An alternative is to do no rescaling at all, i.e. \(\alpha(t) = 1\). This is called variance-exploding or VE diffusion. It requires \(\sigma(t)\) to grow quite large to be able to drown out all of the signal for large values of \(t\), which is a prerequisite for diffusion models to work well. For image pixels scaled to \([-1, 1]\), we might want to ramp up \(\sigma(t)\) all the way to ~100 before it becomes more or less impossible to discern any remaining signal structure. The exact maximum value is a hyperparameter which depends on the data distribution. It was popularised by Karras et al. (2022)².

• More recently, formalisms based on flow matching³ and rectified flow⁴ have gained popularity. They set \(\alpha(t) = 1 - \sigma(t)\), which is also sometimes referred to as sub-VP diffusion. This is because in this case, \(\mathrm{Var}[\mathbf{x}_t] \leq 1\) when we assume \(\mathrm{Var}[\mathbf{x}_0] = 1\). This choice is supposed to result in straighter paths through input space between data and noise, which in turn reduces the number of sampling steps required to hit a certain level of quality (see my previous blog post for more about sampling with fewer steps). Stable Diffusion 3 uses this approach⁵.

By convention, \(t\) typically ranges from \(0\) to \(1\) in the VP and sub-VP settings, so that no noise is present at \(t=0\) (hence \(\sigma(0) = 0\) and \(\alpha(0) = 1\)), and at \(t=1\) the noise has completely drowned out the signal (hence \(\sigma(1) = 1\) and \(\alpha(1) = 0\)). In the flow matching literature, the direction of \(t\) is usually reversed, so that \(t=0\) corresponds to maximal noise and \(t=1\) to minimal noise instead, but I am sticking to the diffusion convention here. Note that \(t\) can be a continuous time variable, or a discrete index, depending on which paper you’re reading; here, we will assume it is continuous.

Let’s look at a few different noise schedules that have been used in the literature. It goes without saying that this is far from an exhaustive list – I will only mention some of the most popular and interesting options.

• The so-called linear schedule was proposed in the original DDPM paper⁶. This paper uses a discrete-time formulation, and specifies the schedule in terms of the variances of \(q(\mathbf{x}_{t+1} \mid \mathbf{x}_t)\) (corresponding to a single discrete step in the forward process), which they call \(\beta_t\). These variances increase linearly with \(t\), which is where the name comes from. In our formalism, this corresponds to \(\sigma(t) = \sqrt{1 - \prod_{i=1}^t (1 - \beta_i)}\), so while \(\beta_t\) might be a linear function of \(t\), \(\sigma(t)\) is not.

• The cosine schedule is arguably the most popular noise schedule to this day. It was introduced by Nichol & Dhariwal⁷ after observing that the linear schedule is suboptimal for high-resolution images, because it gets too noisy too quickly. This corresponds to \(\sigma(t) = \sin \left(\frac{t/T + s}{1 + s} \frac{\pi}{2} \right)\), where \(T\) is the maximal (discrete) time step, and \(s\) is an offset hyperparameter. It might seem like calling this the sine schedule would have been more appropriate, but the naming is again the result of using a slightly different formalism. (There is no standardised formalism for diffusion models, so every paper tends to describe things using different conventions and terminology, which is something I’ve written about before.)

• Karras et al. (2022)² use the variance-exploding formalism in combination with the simplest noise schedule you can imagine: \(\sigma(t) = t\). Because of this, they get rid of the “time” variable altogether, and express everything directly in terms of \(\sigma\) (because they are effectively equivalent). This is not the whole story however, and we’ll revisit this approach later.

• To adjust a pre-existing noise schedule to be more suitable for high-resolution images, both Chen (2023)⁸ and Hoogeboom et al. (2023)⁹ suggest “shifting” the schedule to account for the fact that neighbouring pixels in high-resolution images exhibit much stronger correlations than in low-resolution images, so more noise is needed to obscure any structure that is present. They do this by expressing the schedule in terms of the signal-to-noise ratio, \(\mathrm{SNR}(t) = \frac{\alpha(t)^2}{\sigma(t)^2}\), and showing that halving the resolution along both the width and height dimensions (dividing the total number of pixels by 4) requires scaling \(SNR(t)\) by a factor of 4 to ensure the same level of corruption at time \(t\). If we express the noise schedule in terms of the logarithm of the SNR, this means we simply have to additively shift the input by \(\log 4\), or by \(- \log 4\) when doubling the resolution instead.

There is a monotonically decreasing (and hence, invertible) relationship between the time variable of the diffusion process and the logSNR. Representing things in terms of the logSNR instead of time is quite useful: it is a direct measure of the amount of information obscured by noise, and is therefore easier to compare across different settings: different models, different noise schedules, but also across VP, VE and sub-VP formulations.

Noise levels: focusing on what matters

Let’s dive a bit deeper into the role that noise schedules fulfill. Compared to other classes of generative models, diffusion models have a superpower: because they generate things step-by-step in a coarse-to-fine or hierarchical manner, we can determine which levels of this hierarchy are most important to us, and use the bulk of their capacity for those. There is a very close correspondence between noise levels and levels of the hierarchy.

This enables diffusion models to be quite compute- and parameter-efficient for perceptual modalities in particular: sound, images and video exhibit a huge amount of variation in relative importance across different levels of granularity, with respect to perceptual quality. More concretely, human eyes and ears are much more sensitive to low frequencies than high frequencies, and diffusion models can exploit this out of the box by spending more effort on modelling lower frequencies, which correspond to higher noise levels. (Incidentally, I believe this is one of the reasons why they haven’t really caught on for language modelling, where this advantage does not apply – I have a blog post about that as well.)

In what follows, I will focus on these perceptual use cases, but the observations and conclusions are also applicable to diffusion models of other modalities. It’s just convenient to talk about perceptual quality as a stand-in for “aspects of sample quality that we care about”.

So which noise levels should we focus on when training a diffusion model, and how much? I believe the two most important matters that affect this decision are:

• the perceptual relevance of each noise level, as previously discussed;
• the difficulty of the learning task at each noise level.

Neither of these are typically uniformly distributed. It’s also important to consider that these distributions are not necessarily similar to each other: a noise level that is highly relevant perceptually could be quite easy for the model to learn to make predictions for, and vice versa. Noise levels that are particularly difficult could be worth focusing on to improve output quality, but they could also be so difficult as to be impossible to learn, in which case any effort expended on them would be wasted.

To find the optimal balance between noise levels during model training, we need to take both perceptual relevance and difficulty into account. This always comes down to a trade-off between different priorities: model capacity is finite, and focusing training on certain noise levels will necessarily reduce a model’s predictive capability at other noise levels.

When sampling from a trained diffusion model, the situation is a bit different. Here, we need to choose how to space things out as we traverse the different noise levels from high to low. In a range of noise levels that is more important, we’ll want to spend more time evaluating the model, and therefore space the noise levels closer together. As the number of sampling steps we can afford is usually limited, this means we will have to space the noise levels farther apart elsewhere. The importance of noise levels during sampling is affected by:

• their perceptual relevance, as is the case for model training;
• the accuracy of model predictions;
• the possibility for accumulation of errors.

While prediction accuracy is of course closely linked to the difficulty of the learning task, it is not the same thing. The accumulation of errors over the course of the sampling process also introduces an asymmetry, as errors made early in the process (at high noise levels) are more likely to lead to problems than those made later on (at low noise levels). These subtle differences can result in an optimal balance between noise levels that looks very different than at training time, as we will see later.

Model design choices: what might tip the balance?

Now that we have an idea of what affects the relative importance of noise levels, both for training and sampling, we can analyse the various design choices we need to make when constructing a diffusion model, and how they influence this balance. As it turns out, the choice of noise schedule is far from the only thing that matters.

A good starting point is to look at how we estimate the training loss:

Here, \(p(\mathbf{x}_0)\) is the data distribution, and \(p(\mathbf{x}_t \mid \mathbf{x}_0, t)\) represents the so-called transition density of the forward diffusion process, which describes the distribution of the noisy input \(\mathbf{x}_t\) at time step \(t\) if we started the corruption process at a particular training example \(\mathbf{x}_0\) at \(t = 0\). In addition to the noise schedule \(\sigma(t)\), there are three aspects of the loss that together determine the relative importance of noise levels: the model output parameterisation \(\color{purple}{f(\mathbf{x}_t, t)}\), the loss weighting \(\color{blue}{w(t)}\) and the time step distribution \(\color{red}{p(t)}\). We’ll take a look at each of these in turn.

Model output parameterisation \(\color{purple}{f(\mathbf{x}_t, t)}\)

For a typical diffusion model, we sample from the transition density in practice by sampling standard Gaussian noise \(\varepsilon \sim \mathcal{N}(0, 1)\) and constructing \(\mathbf{x}_t = \alpha(t) \mathbf{x}_0 + \sigma(t) \varepsilon\), i.e. a weighted mix of the data distribution and standard Gaussian noise, with \(\sigma(t)\) the noise schedule and \(\alpha(t)\) defined accordingly (see Section 1). This implies that the transition density is Gaussian: \(p(\mathbf{x}_t \mid \mathbf{x}_0, t) = \mathcal{N}(\alpha(t) \mathbf{x}_0, \sigma(t)^2)\).

Here, we have chosen to parameterise the model \(\color{purple}{f(\mathbf{x}_t, t)}\) to predict the corresponding clean input \(\mathbf{x}_0\), following Karras et al.². This is not the only option: it is also common to have the model predict \(\varepsilon\), or a linear combination of the two, which can be time-dependent (as in \(\mathbf{v}\)-prediction¹⁰, \(\mathbf{v} = \alpha(t) \varepsilon - \sigma(t) \mathbf{x}_0\), or as in rectified flow⁴, where the target is \(\varepsilon - \mathbf{x}_0\)).

Once we have a prediction \(\hat{\mathbf{x}}_0 = \color{purple}{f(\mathbf{x}_t, t)}\), we can easily turn this into a prediction \(\hat{\varepsilon}\) or \(\hat{\mathbf{v}}\) corresponding to a different parameterisation, using the linear relation \(\mathbf{x}_t = \alpha(t) \mathbf{x}_0 + \sigma(t) \varepsilon\), because \(t\) and \(\mathbf{x}_t\) are given. You would be forgiven for thinking that this implies all of these parameterisations are essentially equivalent, but that is not the case.

Depending on the choice of parameterisation, different noise levels will be emphasised or de-emphasised in the loss, which is an expectation across all time steps. To see why, consider the expression \(\mathbb{E}[(\hat{\mathbf{x}_0} - \mathbf{x}_0)^2]\), i.e. the mean squared error w.r.t. the clean input \(\mathbf{x}_0\), which we can rewrite in terms of \(\varepsilon\):

The factor \(\frac{\sigma(t)^2}{\alpha(t)^2}\) which appears in front is the reciprocal of the signal-to-noise ratio \(\mathrm{SNR}(t) = \frac{\alpha(t)^2}{\sigma(t)^2}\). As a result, when we switch our model output parameterisation from predicting \(\mathbf{x}_0\) to predicting \(\varepsilon\) instead, we are implicitly introducing a relative weighting factor equal to \(\mathrm{SNR}(t)\).

We can also rewrite the MSE in terms of \(\mathbf{v}\):

In the VP case, the denominator is equal to \(1\).

These implicit weighting factors will compound with other design choices to determine the relative contribution of each noise level to the overall loss, and therefore, influence the way model capacity is distributed across noise levels. Concretely, this means that a noise schedule tuned to work well for a model that is parameterised to predict \(\mathbf{x}_0\), cannot be expected to work equally well when we parameterise the model to predict \(\varepsilon\) or \(\mathbf{v}\) instead (or vice versa).

This is further complicated by the fact that the model output parameterisation also affects the feasibility of the learning task at different noise levels: predicting \(\varepsilon\) at low noise levels is more or less impossible, so the optimal thing to do is to predict the mean (which is 0). Conversely, predicting \(\mathbf{x}_0\) is challenging at high noise levels, although somewhat more constrained in the conditional setting, where the optimum is to predict the conditional mean across the dataset.

Aside: to disentangle these two effects, one could parameterise the model to predict one quantity (e.g. \(\mathbf{x}_0\)), convert the model predictions to another parameterisation (e.g. \(\varepsilon\)), and express the loss in terms of that, thus changing the implicit weighting. However, this can also be achieved simply by changing \(\color{blue}{w(t)}\) or \(\color{red}{p(t)}\) instead.

Loss weighting \(\color{blue}{w(t)}\)

Many diffusion model formulations feature an explicit time-dependent weighting function in the loss. Karras et al.²’s formulation (often referred to as EDM) features an explicit weighting function \(\lambda(\sigma)\), to compensate for the implicit weighting induced by their choice of parameterisation.

In the original DDPM paper⁶, this weighting function arises from the derivation of the variational bound, but is then dropped to obtain the “simple” loss function in terms of \(\varepsilon\) (§3.4 in the paper). This is found to improve sample quality, in addition to simplifying the implementation. Dropping the weighting results in low noise levels being downweighted considerably compared to high ones, relative to the variational bound. For some applications, keeping this weighting is useful, as it enables training of diffusion models to maximise the likelihood in the input space^{11 12} – lossless compression is one such example.

Time step distribution \(\color{red}{p(t)}\)

During training, a random time step is sampled for each training example \(\mathbf{x}_0\). Most formulations sample time steps uniformly (including DDPM), but some, like EDM² and Stable Diffusion 3⁵, choose a different distribution instead. It stands to reason that this will also affect the balance between noise levels, as some levels will see a lot more training examples than others.

Note that a uniform distribution of time steps usually corresponds to a non-uniform distribution of noise levels, because \(\sigma(t)\) is a nonlinear function. In fact, in the VP case (where \(t, \sigma \in [0, 1]\)), it is precisely the inverse of the cumulative distribution function (CDF) of the resulting noise level distribution.

It turns out that \(\color{blue}{w(t)}\) and \(\color{red}{p(t)}\) are in a sense interchangeable. To see this, simply write out the expectation over \(t\) in the loss as an integral:

It’s pretty obvious now that we are really just multiplying the density of the time step distribution \(\color{red}{p(t)}\) with the weighting function \(\color{blue}{w(t)}\), so we could just absorb \(\color{red}{p(t)}\) into \(\color{blue}{w(t)}\) and make the time step distribution uniform:

Alternatively, we could absorb \(\color{blue}{w(t)}\) into \(\color{red}{p(t)}\) instead. We may have to renormalise it to make sure it is still a valid distribution, but that’s okay, because scaling a loss function by an arbitrary constant factor does not change where the minimum is:

So why would we want to use \(\color{blue}{w(t)}\) or \(\color{red}{p(t)}\), or some combination of both? In practice, we train diffusion models with minibatch gradient descent, which means we stochastically estimate the expectation through sampling across batches of data. The integral over \(t\) is estimated by sampling a different value for each training example. In this setting, the choice of \(\color{red}{p(t)}\) and \(\color{blue}{w(t)}\) affects the variance of said estimate, as well as that of its gradient. For efficient training, we of course want the loss estimate to have the lowest variance possible, and we can use this to inform our choice¹¹.

You may have recognised this as the key idea behind importance sampling, because that’s exactly what this is.

Time step spacing

Once a model is trained and we want to sample from it, \(\color{blue}{w(t)}\), \(\color{red}{p(t)}\) and the choice of model output parameterisation are no longer of any concern. The only thing that determines the relative importance of noise levels at this point, apart from the noise schedule \(\sigma(t)\), is how we space the time steps at which we evaluate the model in order to produce samples.

In most cases, time steps are uniformly spaced (think np.linspace) and not much consideration is given to this. Note that this spacing of time steps usually gives rise to a non-uniform spacing of noise levels, because the noise schedule \(\sigma(t)\) is typically nonlinear.

An exception is EDM², with its simple (linear) noise schedule \(\sigma(t) = t\). Here, the step spacing is intentionally done in a nonlinear fashion, to put more emphasis on lower noise levels. Another exception is the DPM-Solver paper¹³, where the authors found that their proposed fast deterministic sampling algorithm benefits from uniform spacing of noise levels when expressed in terms of logSNR. The latter example demonstrates that the optimal time step spacing can also depend on the choice of sampling algorithm. Stochastic algorithms tend to have better error-correcting properties than deterministic ones, reducing the potential for errors to accumulate over multiple steps².

Noise schedules are a superfluous abstraction

With everything we’ve discussed in the previous two sections, you might ask: what do we actually need the noise schedule for? What role does the “time” variable \(t\) play, when what we really care about is the relative importance of noise levels?

Good question! We can reexpress the loss from the previous section directly in terms of the standard deviation of the noise \(\sigma\):

This is actually quite a straightforward change of variables, because \(\sigma(t)\) is a monotonic and invertible function of \(t\). I’ve also gone ahead and replaced the subscripts \(t\) with \(\sigma\) instead. Note that this is a slight abuse of notation: \(\color{blue}{w(\sigma)}\) and \(\color{blue}{w(t)}\) are not the same functions applied to different arguments, they are actually different functions. The same holds for \(\color{red}{p}\) and \(\color{purple}{f}\). (Adding additional subscripts or other notation to make this difference explicit seemed like a worse option.)

Another possibility is to express everything in terms of the logSNR \(\lambda\):

This is again possible because of the monotonic relationship that exists between \(\lambda\) and \(t\) (and \(\sigma\), for that matter). One thing to watch out for when doing this, is that high logSNRs \(\lambda\) correspond to low standard deviations \(\sigma\), and vice versa.

Once we perform one of these substitutions, the time variable becomes superfluous. This shows that the noise schedule does not actually add any expressivity to our formulation – it is merely an arbitrary nonlinear function that we use to convert back and forth between the domain of time steps and the domain of noise levels. In my opinion, that means we are actually making things more complicated than they need to be.

I’m hardly the first to make this observation: Karras et al. (2022)² figured this out about two years ago, which is why they chose \(\sigma(t) = t\), and then proceeded to eliminate \(t\) everywhere, in favour of \(\sigma\). One might think this is only possible thanks to the variance-exploding formulation they chose to use, but in VP or sub-VP formulations, one can similarly choose to express everything in terms of \(\sigma\) or \(\lambda\) instead.

In addition to complicating things with a superfluous variable and unnecessary nonlinear functions, I have a few other gripes with noise schedules:

• They needlessly entangle the training and sampling importance of noise levels, because changing the noise schedule simultaneously impacts both. This leads to people doing things like using different noise schedules for training and sampling, when it makes more sense to modify the training weighting and sampling spacing of noise levels directly.

• They cause confusion: a lot of people are under the false impression that the noise schedule (and only the noise schedule) is what determines the relative importance of noise levels. I can’t blame them for this misunderstanding, because it definitely sounds plausible based on the name, but I hope it is clear at this point that this is not accurate.

• When combining a noise schedule with uniform time step sampling and uniform time step spacing, as is often done, there is an underlying assumption that specific noise levels are equally important for both training and sampling. This is typically not the case (see Section 2), and the EDM paper also supports this by separately tuning the noise level distribution \(\color{red}{p(\sigma)}\) and the sampling spacing. Kingma & Gao¹⁴ express these choices as weighting functions in terms of the logSNR, demonstrating just how different they end up being (see Figure 2 in their paper).

So do noise schedules really have no role to play in diffusion models? That’s probably an exaggeration. Perhaps they were a necessary concept that had to be invented to get to where we are today. They are pretty key in connecting diffusion models to the theory of stochastic differential equations (SDEs) for example, and seem inevitable in any discrete-time formalism. But for practitioners, I think the concept does more to muddy the waters than to enhance our understanding of what’s going on. Focusing instead on noise levels and their relative importance allows us to tease apart the differences between training and sampling, and to design our models to have precisely the weighting we intended.

This also enables us to cast various formulations of diffusion and diffusion-adjacent models (e.g. flow matching³ / rectified flow⁴, inversion by direct iteration¹⁵, …) as variants of the same idea with different choices of noise level weighting, spacing and scaling. I strongly recommend taking a look at appendix D of Kingma & Gao’s “Understanding diffusion objectives” paper for a great overview of these relationships. In Section 2 and Appendix C of the EDM paper, Karras et al. perform a similar exercise, and this is also well worth reading. The former expresses everything in terms of the logSNR \(\lambda\), the latter uses the standard deviation \(\sigma\).

Adaptive weighting mechanisms

A few heuristics and mechanisms to automatically balance the importance of different noise levels have been proposed in the literature, both for training and sampling. I think this is a worthwhile pursuit, because optimising what is essentially a function-valued hyperparameter can be quite costly and challenging in practice. For some reason, these ideas are frequently tucked away in the appendices of papers that make other important contributions as well.

• The “Variational Diffusion Models” paper¹¹ uses a fixed noise level weighting for training, corresponding to the likelihood loss (or rather, a variational bound on it). But as we discussed earlier, given a particular choice of model output parameterisation, any weighting can be implemented either through an explicit weighting factor \(\color{blue}{w(t)}\), a non-uniform time step distribution \(\color{red}{p(t)}\), or some combination of both, which affects the variance of the loss estimate. They show how this variance can be minimised explicitly by parameterising the noise schedule with a neural network, and optimising its parameters to minimise the squared diffusion loss, alongside the denoising model itself (see Appendix I.2). This idea is also compatible with other choices of noise level weighting.

• The “Understanding Diffusion Objectives” paper¹⁴ proposes an alternative online mechanism to reduce variance. Rather than minimising the variance directly, expected loss magnitude estimates are tracked across a range of logSNRs divided into a number of discrete bins, by updating an exponential moving average (EMA) after every training step. These are used for importance sampling: we can construct an adaptive piecewise constant non-uniform noise level distribution \(\color{red}{p(\lambda)}\) that is proportional to these estimates, which means noise levels with a higher expected loss value will be sampled more frequently. This is compensated for by multiplying the explicit weighting function \(\color{blue}{w(\lambda)}\) by the reciprocal of \(\color{red}{p(\lambda)}\), which means the effective weighting is kept unchanged (see Appendix F).

• In “Analyzing and Improving the Training Dynamics of Diffusion Models”, also known as the EDM2 paper¹⁶, Karras et al. describe another adaptation mechanism which at first glance seems quite similar to the one above, because it also works by estimating loss magnitudes (see Appendix B.2). There are a few subtle but crucial differences, though. Their aim is to keep gradient magnitudes across different noise levels balanced throughout training. This is achieved by adapting the explicit weighting \(\color{blue}{w(\sigma)}\) over the course of training, instead of modifying the noise level distribution \(\color{red}{p(\sigma)}\) as in the preceding method (here, this is kept fixed throughout). The adaptation mechanism is based on a multi-task learning approach¹⁷, which works by estimating the loss magnitudes across noise levels with a one-layer MLP, and normalising the loss contributions accordingly. The most important difference is that this is not compensated for by adapting \(\color{red}{p(\sigma)}\), so this mechanism actually changes the effective weighting of noise levels over the course of training, unlike the previous two.

• In “Continuous diffusion for categorical data” (CDCD), my colleagues and I developed an adaptive mechanism we called “time warping”¹⁸. We used the categorical cross-entropy loss to train diffusion language models – the same loss that is also used to train autoregressive language models. Time warping tracks the cross-entropy loss values across noise levels using a learnable piecewise linear function. Rather than using this information for adaptive rescaling, the learnt function is interpreted as the (unnormalised) cumulative distribution function (CDF) of \(\color{red}{p(\sigma)}\). Because the estimate is piecewise linear, we can easily normalise it and invert it, enabling us to sample from \(\color{red}{p(\sigma)}\) using inverse transform sampling (\(\color{blue}{w(\sigma)} = 1\) is kept fixed). If we interpret the cross-entropy loss as measuring the uncertainty of the model in bits, the effect of this procedure is to balance model capacity between all bits of information contained in the data.

• In “Continuous Diffusion for Mixed-Type Tabular Data”, Mueller et al.¹⁹ extend the time warping mechanism to heterogeneous data, and use it to learn different noise level distributions \(\color{red}{p(\sigma)}\) for different data types. This is useful in the context of continuous diffusion on embeddings which represent discrete categories, because a given corruption process may destroy the underlying categorical information at different rates for different data types. Adapting \(\color{red}{p(\sigma)}\) to the data type compensates for this, and ensures information is destroyed at the same rate across all data types.

All of the above mechanisms adapt the noise level weighting in some sense, but they vary along a few axes:

• Different aims: minimising the variance of the loss estimate, balancing the magnitude of the gradients, balancing model capacity, balancing corruption rates across heterogeneous data types.
• Different tracking methods: EMA, MLPs, piecewise linear functions.
• Different ways of estimating noise level importance: squared diffusion loss, measuring the loss magnitude directly, multi-task learning, fitting the CDF of \(\color{red}{p(\sigma)}\).
• Different ways of employing this information: it can be used to adapt \(\color{red}{p}\) and \(\color{blue}{w}\) together, only \(\color{red}{p}\), or only \(\color{blue}{w}\). Some mechanisms change the effective weighting \(\color{red}{p} \cdot \color{blue}{w}\) over the course of training, others keep it fixed.

Apart from these online mechanisms, which adapt hyperparameters on-the-fly over the course of training, one can also use heuristics to derive weightings offline that are optimal in some sense. Santos & Lin (2023) explore this setting, and propose four different heuristics to obtain noise schedules for continuous variance-preserving Gaussian diffusion²⁰. One of them, based on the Fisher Information, ends up recovering the cosine schedule. This is a surprising result, given its fairly ad-hoc origins. Whether there is a deeper connection here remains to be seen, as this derivation does not account for the impact of perceptual relevance on the relative importance of noise levels, which I think plays an important role in the success of the cosine schedule.

The mechanisms discussed so far apply to model training. We can also try to automate finding the optimal sampling step spacing for a trained model. A recent paper titled “Align your steps”²¹ proposes to optimise the spacing by analytically minimising the discretisation error that results from having to use finite step sizes. For smaller step budgets, some works have treated the individual time steps as sampling hyperparameters that can be optimised via parameter sweeping or black-box optimisation: the WaveGrad paper²² is an example where a high-performing schedule with only 6 steps was found in this way.

In CDCD, we found that reusing the learnt CDF of \(\color{red}{p(\sigma)}\) to also determine the sampling spacing of noise levels worked very well in practice. This seemingly runs counter to the observation made in the EDM paper², that optimising the sampling spacing separately from the training weighting is worthwhile. My current hypothesis for this is as follows: in the language domain, information is already significantly compressed, to such an extent that every bit ends up being roughly equally important for output quality and performance on downstream tasks. (This also explains why balancing model capacity across all bits during training works so well in this setting.) We know that this is not the case at all for perceptual signals such as images: for every perceptually meaningful bit of information in an uncompressed image, there are 99 others that are pretty much irrelevant (which is why lossy compression algorithms such as JPEG are so effective).

Closing thoughts

I hope I have managed to explain why I am not a huge fan of the noise schedule as a central abstraction in diffusion model formalisms. The balance between different noise levels is determined by much more than just the noise schedule: the model output parameterisation, the explicit time-dependent weighting function (if any), and the distribution which time steps are sampled from all have a significant impact during training. When sampling, the spacing of time steps also plays an important role.

All of these should be chosen in tandem to obtain the desired relative weighting of noise levels, which might well be different for training and sampling, because the optimal weighting in each setting is affected by different things: the difficulty of the learning task at each noise level (training), the accuracy of model predictions (sampling), the possibility for error accumulation (sampling) and the perceptual relevance of each noise level (both). An interesting implication of this is that finding the optimal weightings for both settings actually requires bilevel optimisation, with an outer loop optimising the training weighting, and an inner loop optimising the sampling weighting.

As a practitioner, it is worth being aware of how all these things interact, so that changing e.g. the model output parameterisation does not lead to a surprise drop in performance, because the accompanying implicit change in the relative weighting of noise levels was not accounted for. The “noise schedule” concept unfortunately creates the false impression that it solely determines the relative importance of noise levels, and needlessly entangles them across training and sampling. Nevertheless, it is important to understand the role of noise schedules, as they are pervasive in the diffusion literature.

Two papers were instrumental in developing my own understanding: the EDM paper² (yes, I am aware that I’m starting to sound like a broken record!) and the “Understanding diffusion objectives” paper¹⁴. They are both really great reads (including the various appendices), and stuffed to the brim with invaluable wisdom. In addition, the recent Stable Diffusion 3 paper⁵ features a thorough comparison study of different noise schedules and model output parameterisations.

I promised I would explain the title: this is of course a reference to Dijkstra’s famous essay about the “go to” statement. It is perhaps the most overused of all snowclones in technical writing, but I chose it specifically because the original essay also criticised an abstraction that sometimes does more harm than good.

This blog post took a few months to finish, including several rewrites, because the story is quite nuanced. The precise points I wanted to make didn’t become clear even to myself, until about halfway through writing it, and my thinking on this issue is still evolving. If anything is unclear (or wrong!), please let me know. I am curious to learn if there are any situations where an explicit time variable and/or a noise schedule simplifies or clarifies things, which would not be obvious when expressed directly in terms of the standard deviation \(\sigma\), or the logSNR \(\lambda\). I also want to know about any other adaptive mechanisms that have been tried. Let me know in the comments, or come find me at ICML 2024 in Vienna!

If you would like to cite this post in an academic context, you can use this BibTeX snippet:

@misc{dieleman2024schedules,
  author = {Dieleman, Sander},
  title = {Noise schedules considered harmful},
  url = {https://sander.ai/2024/06/14/noise-schedules.html},
  year = {2024}
}

Acknowledgements

Thanks to Robin Strudel, Edouard Leurent, Sebastian Flennerhag and all my colleagues at Google DeepMind for various discussions, which continue to shape my thoughts on diffusion models and beyond!

References

Diffusion models split up the difficult task of generating data from a high-dimensional distribution into many denoising tasks, each of which is much easier. We train them to solve just one of these tasks at a time. To sample, we make many predictions in sequence. This iterative refinement is where their power comes from.
…or is it? A lot of recent papers about diffusion models focus on reducing the number of sampling steps required; some works even aim to enable single-step sampling. That seems counterintuitive, when splitting things up into many easier steps is supposedly why these models work so well in the first place!

In this blog post, let’s take a closer look at the various ways in which the number of sampling steps required to get good results from diffusion models can be reduced. We will focus on various forms of distillation in particular: this is the practice of training a new model (the student) by supervising it with the predictions of another model (the teacher). Various distillation methods for diffusion models have produced extremely compelling results.

I intended this to be relatively high-level when I started writing, but since distillation of diffusion models is a bit of a niche subject, I could not avoid explaining certain things in detail, so it turned into a deep dive. Below is a table of contents. Click to jump directly to a particular section of this post.

1. Diffusion sampling: tread carefully!
2. Moving through input space with purpose
3. Diffusion distillation
   • Distilling diffusion sampling into a single forward pass
   • Progressive distillation
   • Guidance distillation
   • Rectified flow
   • Consistency distillation & TRACT
   • BOOT: data-free distillation
   • Sampling with neural operators
   • Score distillation sampling
   • Adversarial distillation
4. But what about “no free lunch”?
5. Do we really need a teacher?
6. Charting the maze between data and noise
7. Closing thoughts
8. Acknowledgements
9. References

Diffusion sampling: tread carefully!

First of all, why does it take many steps to get good results from a diffusion model? It’s worth developing a deeper understanding of this, in order to appreciate how various methods are able to cut down on this without compromising the quality of the output – or at least, not too much.

A sampling step in a diffusion model consists of:

• predicting the direction in input space in which we should move to remove noise, or equivalently, to make the input more likely under the data distribution;
• taking a small step in that direction.

Depending on the sampling algorithm, you might add a bit of noise, or use a more advanced mechanism to compute the update direction.

We only take a small step, because this predicted direction is only meaningful locally: it points towards a region of input space where the likelihood under the data distribution is high – not to any specific data point in particular. So if we were to take a big step, we would end up in the centroid of that high-likelihood region, which isn’t necessarily a representative sample of the data distribution. Think of it as a rough estimate. If you find this unintuitive, you are not alone! Probability distributions in high-dimensional spaces often behave unintuitively, something I’ve written an an in-depth blog post about in the past.

Concretely, in the image domain, taking a big step in the predicted direction tends to yield a blurry image, if there is a lot of noise in the input. This is because it basically corresponds to the average of many plausible images. (For the sake of argument, I am intentionally ignoring any noise that might be added back in as part of the sampling algorithm.)

Another way of looking at it is that the noise obscures high-frequency information, which corresponds to sharp features and fine-grained details (something I’ve also written about before). The uncertainty about this high-frequency information yields a prediction where all the possibilities are blended together, which results in a lack of high-frequency information altogether.

The local validity of the predicted direction implies we should only be taking infinitesimal steps, and then reevaluating the model to determine a new direction. Of course, this is not practical, so we take finite but small steps instead. This is very similar to the way gradient-based optimisation of machine learning models works in parameter space, but here we are operating in the input space instead. Just as in model training, if the steps we take are too large, the quality of the end result will suffer.

Below is a diagram that represents the input space in two dimensions. \(\mathbf{x}_t\) represents the noisy input at time step \(t\), which we constructed here by adding noise to a clean image \(\mathbf{x}_0\) drawn from the data distribution. Also shown is the direction (predicted by a diffusion model) in which we should move to make the input more likely. This points to \(\hat{\mathbf{x}}_0\), the centroid of a region of high likelihood, which is shaded in pink.

(Please see the first section of my previous blog post on the geometry of diffusion guidance for some words of caution about representing very high-dimensional spaces in 2D!)

If we proceed to take a step in this direction and add some noise (as we do in the DDPM¹ sampling algorithm, for example), we end up with \(\mathbf{x}_{t-1}\), which corresponds to a slightly less noisy input image. The predicted direction now points to a smaller, “more specific” region of high likelihood, because some uncertainty was resolved by the previous sampling step. This is shown in the diagram below.

The change in direction at every step means that the path we trace out through input space during sampling is curved. Actually, because we are making a finite approximation, that’s not entirely accurate: it is actually a piecewise linear path. But if we let the number of steps go to infinity, we would end up with a curve. The predicted direction at each point on this curve corresponds to the tangent direction. A stylised version of what this curve might look like is shown in the diagram below.

Moving through input space with purpose

A plethora of diffusion sampling algorithms have been developed to move through input space more swiftly and reduce the number of sampling steps required to achieve a certain level of output quality. Trying to list all of them here would be a hopeless endeavour, but I want to highlight a few of these algorithms to demonstrate that a lot of the ideas behind them mimic techniques used in gradient-based optimisation.

A very common question about diffusion sampling is whether we should be injecting noise at each step, as in DDPM¹, and sampling algorithms based on stochastic differential equation (SDE) solvers². Karras et al.³ study this question extensively (see sections 3 & 4 in their “instant classic” paper) and find that the main effect of introducing stochasticity is error correction: diffusion model predictions are approximate, and noise helps to prevent these approximation errors from accumulating across many sampling steps. In the context of optimisation, the regularising effect of noise in stochastic gradient descent (SGD) is well-studied, so perhaps this is unsurprising.

However, for some applications, injecting randomness at each sampling step is not acceptable, because a deterministic mapping between samples from the noise distribution and samples from the data distribution is necessary. Sampling algorithms such as DDIM⁴ and ODE-based approaches² make this possible (I’ve previously written about this feat of magic, as well as how this links together diffusion models and flow-based models). An example of where this comes in handy is for teacher models in the context of distillation (see next section). In that case, other techniques can be used to reduce approximation error while avoiding an increase in the number of sampling steps.

One such technique is the use of higher order methods. Heun’s 2nd order method for solving differential equations results in an ODE-based sampler that requires two model evaluations per step, which it uses to obtain improved estimates of update directions⁵. While this makes each sampling step approximately twice as expensive, the trade-off can still be favourable in terms of the total number of function evaluations³.

Another variant of this idea involves making the model predict higher-order score functions – think of this as the model estimating both the direction and the curvature, for example. These estimates can then be used to move faster in regions of low curvature, and slow down appropriately elsewhere. GENIE⁶ is one such method, which involves distilling the expensive second order gradient calculation into a small neural network to reduce the additional cost to a practical level.

Finally, we can emulate the effect of higher-order information by aggregating information across sampling steps. This is very similar to the use of momentum in gradient-based optimisation, which also enables acceleration and deceleration depending on curvature, but without having to explicitly estimate second order quantities. In the context of differential equation solving, this approach is usually termed a multistep method, and this idea has inspired many diffusion sampling algorithms^{7 8 9 10}.

In addition to the choice of sampling algorithm, we can also choose how to space the time steps at which we compute updates. These are spaced uniformly across the entire range by default (think np.linspace), but because noise schedules are often nonlinear (i.e. \(\sigma_t\) is a nonlinear function of \(t\)), the corresponding noise levels are spaced in a nonlinear fashion as a result. However, it can pay off to treat sampling step spacing as a hyperparameter to tune separately from the choice of noise schedule (or, equivalently, to change the noise schedule at sampling time). Judiciously spacing out the time steps can improve the quality of the result at a given step budget³.

Diffusion distillation

Broadly speaking, in the context of neural networks, distillation refers to training a neural network to mimic the outputs of another neural network¹¹. The former is referred to as the student, while the latter is the teacher. Usually, the teacher has been trained previously, and its weights are frozen. When applied to diffusion models, something interesting happens: even if the student and teacher networks are identical in terms of architecture, the student will converge significantly faster than the teacher did when it was trained.

To understand why this happens, consider that diffusion model training involves supervising the network with examples \(\mathbf{x}_0\) from the dataset, to which we have added varying amounts of noise to create the network input \(\mathbf{x}_t\). But rather than expecting the network to be able to predict \(\mathbf{x}_0\) exactly, what we actually want is for it to predict \(\mathbb{E}\left[\mathbf{x}_0 \mid \mathbf{x}_t \right]\), that is, a conditional expectation over the data distribution. It’s worth revisiting the first diagram in section 1 of this post to grasp this: we supervise the model with \(\mathbf{x}_0\), but this is not what we want the model to predict – what we actually want is for it to predict a direction pointing to the centroid of a region of high likelihood, which \(\mathbf{x}_0\) is merely a representative sample of. I’ve previously mentioned this when discussing various perspectives on diffusion. This means that weight updates are constantly pulling the model weights in different directions as training progresses, slowing down convergence.

When we distill a diffusion model, rather than training it from scratch, the teacher provides an approximation of \(\mathbb{E}\left[\mathbf{x}_0 \mid \mathbf{x}_t \right]\), which the student learns to mimic. Unlike before, the target used to supervise the model is now already an (approximate) expectation, rather than a single representative sample. As a result, the variance of the distillation loss is significantly reduced compared to that of the standard diffusion training loss. Whereas the latter tends to produce training curves that are jumping all over the place, distillation provides a much smoother ride. This is especially obvious when you plot both training curves side by side. Note that this variance reduction does come at a cost: since the teacher is itself an imperfect model, we’re actually trading variance for bias.

Variance reduction alone does not explain why distillation of diffusion models is so popular, however. Distillation is also a very effective way to reduce the number of sampling steps required. It seems to be a lot more effective in this regard than simply changing up the sampling algorithm, but of course there is also a higher upfront cost, because it requires additional model training.

There are many variants of diffusion distillation, a few of which I will try to compactly summarise below. It goes without saying that this is not an exhaustive review of the literature. A relatively recent survey paper is Weijian Luo’s (from April 2023)¹², though a lot of work has appeared in this space since then, so I will try to cover some newer things as well. If you feel there is a particular method that’s worth mentioning but that I didn’t cover, let me know in the comments.

Distilling diffusion sampling into a single forward pass

A typical diffusion sampling procedure involves repeatedly applying a neural network on a canvas, and using the prediction to update that canvas. When we unroll the computational graph of this network, this can be reinterpreted as a much deeper neural network in its own right, where many layers share weights. I’ve previously discussed this perspective on diffusion in more detail.

Distillation is often used to compress larger networks into smaller ones, so Luhman & Luhman¹³ set out to train a much smaller student network to reproduce the outputs of this much deeper teacher network corresponding to an unrolled sampling procedure. In fact, what they propose is to distill the entire sampling procedure into a network with the same architecture used for a single diffusion prediction step, by matching outputs in the least-squares sense (MSE loss). Depending on how many steps the sampling procedure has, this may correspond to quite an extreme form of model compression (in the sense of compute, that is – the number of parameters stays the same, of course).

This approach requires a deterministic sampling procedure, so they use DDIM⁴ – a choice which many distillation methods that were developed later also follow. The result of their approach is a compact student network which transforms samples from the noise distribution into samples from the data distribution in a single forward pass.

Putting this into practice, one encounters a significant hurdle, though: to obtain a single training example for the student, we have to run the full diffusion sampling procedure using the teacher, which is usually too expensive to do on-the-fly during training. Therefore the dataset for the student has to be pre-generated offline. This is still expensive, but at least it only has to be done once, and the resulting training examples can be reused for multiple epochs.

To speed up the learning process, it also helps to initialise the student with the weights of the teacher (which we can do because their architectures are identical). This is a trick that most diffusion distillation methods make use of.

This work served as a compelling proof-of-concept for diffusion distillation, but aside from the computational cost, the accumulation of errors in the deterministic sampling procedure, combined with the approximate nature of the student predictions, imposed significant limits on the achievable output quality.

Progressive distillation

Progressive distillation¹⁴ is an iterative approach that halves the number of required sampling steps. This is achieved by distilling the output of two consecutive sampling steps into a single forward pass. As with the previous method, this requires a deterministic sampling method (the paper uses DDIM), as well as a predetermined number of sampling steps \(N\) to use for the teacher model.

To reduce the number of sampling steps further, it can be applied repeatedly. In theory, one can go all the way down to single-step sampling by applying the procedure \(\log_2 N\) times. This addresses several shortcomings of the previous approach:

• At each distillation stage, only two consecutive sampling steps are required, which is significantly cheaper than running the whole sampling procedure end-to-end. Therefore it can be done on-the-fly during training, and pre-generating the training dataset is no longer required.
• The original training dataset used for the teacher model can be reused, if it is available (or any other dataset!). This helps to focus learning on the part of input space that is relevant and interesting.
• While we could go all the way down to 1 step, the iterative nature of the procedure enables a trade-off between quality and compute cost. Going down to 4 or 8 steps turns out to help a lot to keep the inevitable quality loss from distillation at bay, while still speeding up sampling very significantly. This also provides a much better trade-off than simply reducing the number of sampling steps for the teacher model, instead of distilling it (see Figure 4 in the paper).

Aside: v-prediction

The most common parameterisation for training diffusion models in the image domain, where the neural network predicts the standardised Gaussian noise variable \(\varepsilon\), causes problems for progressive distillation. The implicit relative weighting of noise levels in the MSE loss w.r.t. \(\varepsilon\) is particularly suitable for visual data, because it maps well to the human visual system’s varying sensitivity to low and high spatial frequencies. This is why it is so commonly used.

To obtain a prediction in input space \(\hat{\mathbf{x}}_0\) from a model that predicts \(\varepsilon\) from the noisy input \(\mathbf{x}_t\), we can use the following formula:

Here, \(\sigma_t\) represents the standard deviation of the noise at time step \(t\). (For variance-preserving diffusion, the scale factor \(\alpha_t = \sqrt{1 - \sigma_t^2}\), for variance-exploding diffusion, \(\alpha_t = 1\).)

At high noise levels, \(\mathbf{x}_t\) is dominated by noise, so the difference between \(\mathbf{x}_t\) and the scaled noise prediction is potentially quite small – but this difference entirely determines the prediction in input space \(\hat{\mathbf{x}}_0\)! This means any prediction errors may get amplified. In standard diffusion models, this is not a problem in practice, because errors can be corrected over many steps of sampling. In progressive distillation, this becomes a problem in later iterations, where we mainly evaluate the model at high noise levels (in the limit of a single-step model, the model is only ever evaluated at the highest noise level).

It turns out this issue can be addressed simply by parameterising the model to predict \(\mathbf{x}_0\) instead, but the progressive distillation paper also introduces a new prediction target \(\mathbf{v} = \alpha_t \varepsilon - \sigma_t \mathbf{x}_0\) (“velocity”, see section 4 and appendix D). This has some really nice properties, and has also become quite popular beyond just distillation applications in recent times.

Guidance distillation

Before moving on to more advanced diffusion distillation methods that reduce the number of sampling steps, it’s worth looking at guidance distillation. The goal of this method is not to achieve high-quality samples in fewer steps, but rather to make each step computationally cheaper when using classifier-free guidance¹⁵. I have already dedicated two entire blog posts specifically to diffusion guidance, so I will not recap the concept here. Check them out first if you’re not familiar:

• Guidance: a cheat code for diffusion models
• The geometry of diffusion guidance

The use of classifier-free guidance requires two model evaluations per sampling step: one conditional, one unconditional. This makes sampling roughly twice as expensive, as the main cost is in the model evaluations. To avoid paying that cost, we can distill predictions that result from guidance into a model that predicts them directly in a single forward pass, conditioned on the chosen guidance scale¹⁶.

While guidance distillation does not reduce the number of sampling steps, it roughly halves the required computation per step, so it still makes sampling roughly twice as fast. It can also be combined with other forms of distillation. This is useful, because reducing the number of sampling steps actually reduces the impact of guidance, which relies on repeated small adjustments to update directions to work. Applying guidance distillation before another distillation method can help ensure that the original effect is preserved as the number of steps is reduced.

Rectified flow

One way to understand the requirement for diffusion sampling to take many small steps, is through the lens of curvature: we can only take steps in a straight line, so if the steps we take are too large, we end up “falling off” the curve, leading to noticeable approximation errors.

As mentioned before, some sampling algorithms compensate for this by using curvature information to determine the step size, or by injecting noise to reduce error accumulation. The rectified flow method¹⁷ takes a more drastic approach: what if we just replace these curved paths between samples from the noise and data distributions with another set of paths that are significantly less curved?

This is possible using a procedure that resembles distillation, though it doesn’t quite have the same goal: whereas distillation tries to learn better/faster approximations of existing paths between samples from the noise and data distributions, the reflow procedure replaces the paths with a new set of paths altogether. We get a new model that gives rise to a set of paths with a lower cost in the “optimal transport” sense. Concretely, this means the paths are less curved. They will also typically connect different pairs of samples than before. In some sense, the mapping from noise to data is “rewired” to be more straight.

Lower curvature means we can take fewer, larger steps when sampling from this new model using our favourite sampling algorithm, while still keeping the approximation error at bay. But aside from that, this also greatly increases the efficacy of distillation, presumably because it makes the task easier.

The procedure can be applied recursively, to yield and even straighter set of paths. After an infinite number of applications, the paths should be completely straight. In practice, this only works up to a certain point, because each application of the procedure yields a new model which approximates the previous, so errors can quickly accumulate. Luckily, only one or two applications are needed to get paths that are mostly straight.

This method was successfully applied to a Stable Diffusion model¹⁸ and followed by a distillation step using a perceptual loss¹⁹. The resulting model produces reasonable samples in a single forward pass. One downside of the method is that each reflow step requires the generation of a dataset of sample pairs (data and corresponding noise) using a deterministic sampling algorithm, which usually needs to be done offline to be practical.

Consistency distillation & TRACT

As we covered before, diffusion sampling traces a curved path through input space, and at each point on this curve, the diffusion model predicts the tangent direction. What if we had a model that could predict the endpoint of the path on the side of the data distribution instead, allowing us to jump there from anywhere on the path in one step? Then the degree of curvature simply wouldn’t matter.

This is what consistency models²⁰ do. They look very similar to diffusion models, but they predict a different kind of quantity: an endpoint of the path, rather than a tangent direction. In a sense, diffusion models and consistency models are just two different ways to describe a mapping between noise and data. Perhaps it could be useful to think of consistency models as the “integral form” of diffusion models (or, equivalently, of diffusion models as the “derivative form” of consistency models).

While it is possible to train a consistency model from scratch (though not that straightforward, in my opinion – more on this later), a more practical route to obtaining a consistency model is to train a diffusion model first, and then distill it. This process is called consistency distillation.

It’s worth noting that the resulting model looks quite similar to what we get when distilling the diffusion sampling procedure into a single forward pass. However, that only lets us jump from one endpoint of a path (at the noise side) to the other (at the data side). Consistency models are able to jump to the endpoint on the data side from anywhere on the path.

Learning to map any point on a path to its endpoint requires paired data, so it would seem that we once again need to run the full sampling process to obtain training targets from the teacher model, which is expensive. However, this can be avoided using a bootstrapping mechanism where, in addition to learning from the teacher, the student also learns from itself.

This hinges on the following principle: the prediction of the consistency model along all points on the path should be the same. Therefore, if we take a step along the path using the teacher, the student’s prediction should be unchanged. Let \(f(\mathbf{x}_t, t)\) represent the student (a consistency model), then we have:

where \(\Delta t\) is the step size and \(\mathbf{x}_{t - \Delta t}\) is the result of a sampling step starting from \(\mathbf{x}_t\), with the update direction given by the teacher. The prediction remains consistent along all points on the path, which is where the name comes from. Note that this is not at all true for diffusion models.

Concurrently with the consistency models paper, transitive closure time-distillation (TRACT)²¹ was proposed as an improvement over progressive distilation, using a very similar bootstrapping mechanism. The details of implementation differ, and rather than predicting the endpoint of a path from any point on the path, as consistency models do, TRACT instead divides the range of time steps into intervals, with the distilled model predicting points on paths at the boundaries of those intervals.

Like progressive distillation, this is a procedure that can be repeated with fewer and fewer intervals, to eventually end up with something that looks pretty much the same as a consistency model (when using a single interval that encompasses the entire time step range). TRACT was proposed as an alternative to progressive distillation which requires fewer distillation stages, thus reducing the potential for error accumulation.

It is well-known that diffusion models benefit significantly from weight averaging^{22 23}, so both TRACT and the original formulation of consistency models use an exponential moving average (EMA) of the student’s weights to construct a self-teacher model, which effectively acts as an additional teacher in the distillation process, alongside the diffusion model. That said, a more recent iteration of consistency models²⁴ does not use EMA.

Another strategy to improve consistency models is to use alternative loss functions for distillation, such as a perceptual loss like LPIPS¹⁹, instead of the usual mean squared error (MSE), which we’ve also seen used before with rectified flow¹⁷.

Recent work on distilling a Stable Diffusion model into a latent consistency model²⁵ has yielded compelling results, producing high-resolution images in 1 to 4 sampling steps.

Consistency trajectory models²⁶ are a generalisation of both diffusion models and consistency models, enabling prediction of any point along a path from any other point before it, as well as tangent directions. To achieve this, they are conditioned on two time steps, indicating the start and end positions. When both time steps are the same, the model predicts the tangent direction, like a diffusion model would.

BOOT: data-free distillation

Instead of predicting the endpoint of a path at the data side from any point on that path, as consistency models learn to do, we can try to predict any point on the path from its endpoint at the noise side. This is what BOOT²⁷ does, providing yet another way to describe a mapping between noise and data. Comparing this formulation to consistency models, one looks like the “transpose” of the other (see diagram below). For those of you who remember word2vec²⁸, it reminds me lot of the relationship between the skip-gram and continuous bag-of-words (CBoW) methods!

Just like consistency models, this formulation enables a form of bootstrapping to avoid having to run the full sampling procedure using the teacher (hence the name, I presume): predict \(\mathbf{x}_t = f(\varepsilon, t)\) using the student, run a teacher sampling step to obtain \(\mathbf{x}_{t - \Delta t}\), then train the student so that \(f(\varepsilon, t - \Delta t) \equiv \mathbf{x}_{t - \Delta t}\).

Because the student only ever takes the noise \(\varepsilon\) as input, we do not need any training data to perform distillation. This is also the case when we directly distill the diffusion sampling procedure into a single forward pass – though of course in that case, we can’t avoid running the full sampling procedure using the teacher.

There is one big caveat however: it turns out that predicting \(\mathbf{x}_t\) is actually quite hard to learn. But there is a neat workaround for this: instead of predicting \(\mathbf{x}_t\) directly, we first convert it into a different target using the identity \(\mathbf{x}_t = \alpha_t \mathbf{x}_0 + \sigma_t \varepsilon\). Since \(\varepsilon\) is given, we can rewrite this as \(\mathbf{x}_0 = \frac{\mathbf{x}_t - \sigma_t \varepsilon}{\alpha_t}\), which corresponds to an estimate of the clean input. Whereas \(\mathbf{x}_t\) looks like a noisy image, this single-step \(\mathbf{x}_0\) estimate looks like a blurry image instead, lacking high-frequency content. This is a lot easier for a neural network to predict.

If we see \(\mathbf{x}_t\) as a mixture of signal and noise, we are basically extracting the “signal” component and predicting that instead. We can easily convert such a prediction back to a prediction of \(\mathbf{x}_t\) using the same formula. Just like \(\mathbf{x}_t\) traces a path through input space which can be described by an ODE, this time-dependent \(\mathbf{x}_0\)-estimate does as well. The BOOT authors call the ODE describing this path the signal-ODE.

Unlike in the original consistency models formulation (as well as TRACT), no exponential moving average is used for the bootstrapping procedure. To reduce error accumulation, the authors suggest using a higher-order solver to run the teacher sampling step. Another requirement to make this method work well is an auxiliary “boundary loss”, ensuring the distilled model is well-behaved at \(t = T\) (i.e. at the highest noise level).

Sampling with neural operators

Diffusion sampling with neural operators (DSNO; also known as DFNO, the acronym seems to have changed at some point!)²⁹ works by training a model that can predict an entire path from noise to data given a noise sample in a single forward pass. While the inputs (\(\varepsilon\)) and targets (\(\mathbf{x}_t\) at various \(t\)) are the same as for a BOOT-distilled student model, the latter is only able to produce a single point on the path at a time.

This seems ambitious – how can a neural network predict an entire path at once, from noise all the way to data? The so-called Fourier neural operator (FNO)³⁰ is used to achieve this. By imposing certain architectural constraints, adding temporal convolution layers and making use of the Fourier transform to represent functions of time in frequency space, we obtain a model that can produce predictions for any number of time steps at once.

A natural question is then: why would we actually want to predict the entire path? When sampling, we only really care about the final outcome, i.e. the endpoint of the path at the data side (\(t = 0\)). For BOOT, the point of predicting the other points on the path is to enable the bootstrapping mechanism used for training. But DSNO does not involve any bootstrapping, so what is the point of doing this here?

The answer probably lies in the inductive bias of the temporal convolution layers, combined with the relative smoothness of the paths through input space learnt by diffusion models. Thanks to this architectural prior, training on other points on the path also helps to improve the quality of the predictions at the endpoint on the data side, that is, the only point on the path we actually care about when sampling in a single step. I have to admit I am not 100% confident that this is the only reason – if there is another compelling reason why this works, please let me know!

Score distillation sampling

Score distillation sampling (SDS)³¹ is a bit different from the methods we’ve discussed so far: rather than accelerating sampling by producing a student model that needs fewer steps for high-quality output, this method is aimed at optimisation of parameterised representations of images. This means that it enables diffusion models to operate on other representations of images than pixel grids, even though that is what they were trained on – as long as those representations produce pixel space outputs that are differentiable w.r.t. their parameters³².

As a concrete example of this, SDS was actually introduced to enable text-to-3D. This is achieved through optimisation of Neural Radiance Field (NeRF)³³ representations of 3D models, using a pretrained image diffusion model applied to random 2D projections to control the generated 3D models through text prompts (DreamFusion).

Naively, one could think that simply backpropagating the diffusion loss at various time steps through the pixel space output produced by the parameterised representation should do the trick. This yields gradient updates w.r.t. the representation parameters that minimise the diffusion loss, which should make the pixel space output look more like a plausible image. Unfortunately, this method doesn’t work very well, even when applied directly to pixel representations.

It turns out this is primarily caused by a particular factor in the gradient, which corresponds to the Jacobian of the diffusion model itself. This Jacobian is poorly conditioned for low noise levels. Simply omitting this factor altogether (i.e. replacing it with the identity matrix) makes things work much better. As an added bonus, it means we can avoid having to backpropagate through the diffusion model. All we need is forward passes, just like in regular diffusion sampling algorithms!

After modifying the gradient in a fairly ad-hoc fashion, it’s worth asking what loss function this modified gradient corresponds to. This is actually the same loss function used in probability density distillation³⁴, which was originally developed to distill autoregressive models for audio waveform generation into feedforward models. I won’t elaborate on this connection here, except to mention that it provides an explanation for the mode-seeking behaviour that SDS seems to exhibit. This behaviour often results in pathologies, which require additional regularisation loss terms to mitigate. It was also found that using a high guidance scale for the teacher (a higher value than one would normally use to sample images) helps to improve results.

Noise-free score distillation (NFSD)³⁵ is a variant that modifies the gradient further to enable the use of lower guidance scales, which results in better sample quality and diversity. Variational score distillation sampling (VSD)³⁶ improves over SDS by optimising a distribution over parameterised representations, rather than a point estimate, which also eliminates the need for high guidance scales.

VSD has in turn been used as a component in more traditional diffusion distillation strategies, aimed at reducing the number of sampling steps. A single-step image generator can easily be reinterpreted as a distribution over parameterised representations, which makes VSD readily applicable to this setting, even if it was originally conceived to improve text-to-3D rather than speed up image generation.

Diff-Instruct³⁷ can be seen as such an application, although it was actually published concurrently with VSD. To distill the knowledge from a diffusion model into a single-step feed-forward generator, they suggest minimising the integral KL divergence (IKL), which is a weighted integral of the KL divergence along the diffusion process (w.r.t. time). Its gradient is estimated by contrasting the predictions of the teacher and those of an auxiliary diffusion model which is concurrently trained on generator outputs. This concurrent training gives it a bit of a GAN³⁸ flavour, but note that the generator and the auxiliary model are not adversaries in this case. As with SDS, the gradient of the IKL with respect to the generator parameters only requires evaluating the diffusion model teacher, but not backpropagating through it – though training the auxiliary diffusion model on generator outputs does of course require backpropagation.

Distribution matching distillation (DMD)³⁹ arrives at a very similar formulation from a different angle. Just like in Diff-Instruct, a concurrently trained diffusion model of the generator outputs is used, and its predictions are contrasted against those of the teacher to obtain gradients for the feed-forward generator. This is combined with a perceptual regression loss (LPIPS¹⁹) on paired data from the teacher, which is pre-generated offline. The latter is only applied on a small subset of training examples, making the computational cost of this pre-generation step less prohibitive.

Adversarial distillation

Before diffusion models completely took over in the space of image generation, generative adversarial networks (GANs)³⁸ offered the best visual fidelity, at the cost of mode-dropping: the diversity of model outputs usually does not reflect the diversity of the training data, but at least they look good. In other words, they trade off diversity for quality. On top of that, GANs generate images in a single forward pass, so they are very fast – much faster than diffusion model sampling.

It is therefore unsurprising that some works have sought to combine the benefits of adversarial models and diffusion models. There are many ways to do so: denoising diffusion GANs⁴⁰ and adversarial score matching⁴¹ are just two examples.

A more recent example is UFOGen⁴², which proposes an adversarial finetuning approach for diffusion models that looks a lot like distillation, but actually isn’t distillation, in the strict sense of the word. UFOGen combines the standard diffusion loss with an adversarial loss. Whereas the standard diffusion loss by itself would result in a model that tries to predict the conditional expectation \(\mathbb{E}\left[\mathbf{x}_0 \mid \mathbf{x}_t \right]\), the additional adversarial loss term allows the model to deviate from this and produce less blurry predictions at high noise levels. The result is a reduction in diversity, but it also enables faster sampling. Both the generator and the discriminator are initialised from the parameters of a pre-trained diffusion model, but this pre-trained model is not evaluated to produce training targets, as would be the case in a distillation approach. Nevertheless, it merits inclusion here, as it is intended to achieve the same goal as most of the distillation approaches that we’ve discussed.

Adversarial diffusion distillation⁴³, on the other hand, is a “true” distillation approach, combining score distillation sampling (SDS) with an adversarial loss. It makes use of a discriminator built on top of features from an image representation learning model, DINO⁴⁴, which was previously also used for a purely adversarial text-to-image model, StyleGAN-T⁴⁵. The resulting student model enables single-step sampling, but can also be sampled from with multiple steps to improve the quality of the results. This method was used for SDXL Turbo, a text-to-image system that enables realtime generation – the generated image is updated as you type.

But what about “no free lunch”?

Why is it that we can get these distilled models to produce compelling samples in just a few steps, when diffusion models take tens or hundreds of steps to achieve the same thing? What about “no such thing as a free lunch”?

At first glance, diffusion distillation certainly seems like a counterexample to what is widely considered a universal truth in machine learning, but there is more to it. Up to a point, diffusion model sampling can probably be made more efficient through distillation at no noticeable cost to model quality, but the regime targeted by most distillation methods (i.e. 1-4 sampling steps) goes far beyond that point, and trades off quality for speed. Distillation is almost always “lossy” in practice, and the student cannot be expected to perfectly mimic the teacher’s predictions. This results in errors which can accumulate across sampling steps, or for some methods, across different phases of the distillation process.

What does this trade-off look like? That depends on the distillation method. For most methods, the decrease in model quality directly affects the perceptual quality of the output: samples from distilled models can often look blurry, or the fine-grained details might look sharp but less realistic, which is especially noticeable in images of human faces. The use of adversarial losses based on discriminators, or perceptual loss functions such as LPIPS¹⁹, is intended to mitigate some of this degradation, by further focusing model capacity on signal content that is perceptually relevant.

Some methods preserve output quality and fidelity of high-frequency content to a remarkable degree, but this then usually comes at cost to the diversity of the samples instead. The adversarial methods discussed earlier are a great example of this, as well as methods based on score distillation sampling, which implicitly optimise a mode-seeking loss function.

So if distillation implies a loss of model quality, is training a diffusion model and then distilling it even worthwhile? Why not train a different type of model instead, such as a GAN, which produces a single-step generator out of the box, without requiring distillation? The key here is that distillation provides us with some degree of control over this trade-off. We gain flexibility: we get to choose how many steps we can afford, and by choosing the right method, we can decide exactly how we’re going to cut corners. Do we care more about fidelity or diversity? It’s our choice!

Do we really need a teacher?

Once we have established that diffusion distillation gives us the kind of model that we are after, with the right trade-offs in terms of output quality, diversity and sampling speed, it’s worth asking whether we even needed distillation to arrive at this model to begin with. In a sense, once we’ve obtained a particular model through distillation, that’s an existence proof, showing that such a model is feasible in practice – but it does not prove that we arrived at that model in the most efficient way possible. Perhaps there is a shorter route? Could we train such a model from scratch, and skip the training of the teacher model entirely?

The answer depends on the distillation method. For certain types of models that can be obtained through diffusion distillation, there are indeed alternative training recipes that do not require distillation at all. However, these tend not to work quite as well as the distillation route. Perhaps this is not that surprising: it has long been known that when distilling a large neural network into a smaller one, we can often get better results than when we train that smaller network from scratch¹¹. The same phenomenon is at play here, because we are distilling a sampling procedure with many steps into one with considerably fewer steps. If we look at the computational graphs of these sampling procedures, the former is much “deeper” than the latter, so what we’re doing looks very similar to distilling a large model into a smaller one.

One instance where you have the choice of distillation or training from scratch, is consistency models. The paper that introduced them²⁰ describes both consistency distillation and consistency training. The latter requires a few tricks to work well, including schedules for some of the hyperparameters to create a kind of “curriculum”, so it is arguably a bit more involved than diffusion model training.

Charting the maze between data and noise

One interesting perspective on diffusion model training that is particularly relevant to distillation, is that it provides a way to uncover an optimal transport map between distributions⁴⁶. Through the probability flow ODE formulation², we can see that diffusion models learn a bijection between noise and data, and it turns out that this mapping is approximately optimal in some sense.

This also explains the observation that different diffusion models trained on similar data tend to learn similar mappings: they are all trying to approximate the same optimum! I tweeted (X’ed?) about this a while back:

With all the recent work on distilling diffusion models into single-pass models, I've been thinking a lot about diffusion model training as solving a kind of optimal transport problem🚐 (1/6)

— Sander Dieleman (@sedielem) December 5, 2023

So far, it seems that diffusion model training is the simplest and most effective (i.e. scalable) way we know of to approximate this optimal mapping, but it is not the only way: consistency training represents a compelling alternative strategy. This makes me wonder what other approaches are yet to be discovered, and whether we might be able to find methods that are even simpler than diffusion model training, or more statistically efficient.

Another interesting link between some of these methods can be found by looking more closely at curvature. The paths connecting samples from the noise and data distributions uncovered by diffusion model training tend to be curved. This is why we need many discrete steps to approximate them accurately when sampling.

We discussed a few approaches to sidestep this issue: consistency models^{20 21} avoid it by changing the prediction target of the model, from the tangent direction at the current position to the endpoint of the curve at the data side. Rectified flow¹⁷ instead replaces the curved paths altogether, with a set of paths that are much straighter. But for perfectly straight paths, the tangent direction will actually point to the endpoint! In other words: in the limiting case of perfectly straight paths, consistency models and diffusion models predict the same thing, and become indistinguishable from each other.

Is that observation practically relevant? Probably not – it’s just a neat connection. But I think it’s worthwhile to cultivate a deeper understanding of deterministic mappings between distributions and how to uncover them at scale, as well as the different ways to parameterise them and represent them. I think this is fertile ground for innovations in diffusion distillation, as well as generative modelling through iterative refinement in a broader sense.

Closing thoughts

As I mentioned at the beginning, this was supposed to be a fairly high-level treatment of diffusion distillation, and why there are so many different ways to do it. I ended up doing a bit of a deep dive, because it’s difficult to talk about the connections between all these methods without also explaining the methods themselves. In reading up on the subject and trying to explain things concisely, I actually learnt a lot. If you want to learn about a particular subject in machine learning research (or really anything else), I can heartily recommend writing a blog post about it.

To wrap things up, I wanted to take a step back and identify a few patterns and trends. Although there is a huge variety of diffusion distillation methods, there are clearly some common tricks and ideas that come back frequently:

• Using deterministic sampling algorithms to obtain targets from the teacher is something that almost all methods rely on. DDIM⁴ is popular, but more advanced methods (e.g. higher-order methods) are also an option.
• The parameters of the student network are usually initialised from those of the teacher. This doesn’t just accelerate convergence, for some methods this is essential for them to work at all. We can do this because the architectures of the teacher and student are often identical, unlike in distillation of discriminative models.
• Several methods make use of perceptual losses such as LPIPS¹⁹ to reduce the negative impact of distillation on low-level perceptual quality.
• Bootstrapping, i.e. having the student learn from itself, is a useful trick to avoid having to run the full sampling algorithm to obtain targets from the teacher. Sometimes using the exponential moving average of the student’s parameters is found to help for this, but this isn’t as clear-cut.

Distillation can interact with other modelling choices. One important example is classifier-free guidance¹⁵, which implicitly relies on there being many sampling steps. Guidance operates by modifying the direction in input space predicted by the diffusion model, and the effect of this will inevitably be reduced if only a few sampling steps are taken. For some methods, applying guidance after distillation doesn’t actually make sense anymore, because the student no longer predicts a direction in input space. Luckily guidance distillation¹⁶ can be used to mitigate the impact of this.

Another instance of this is latent diffusion⁴⁷: when applying distillation to a diffusion model trained in latent space, one important question to address is whether the loss should be applied to the latent representation or to pixels. As an example, the adversarial diffusion distillation (ADD) paper⁴³ explicitly suggests calculating the distillation loss in pixel space for improved stability.

The procedure of first solving a problem as well as possible, and then looking for shortcuts that yield acceptable trade-offs, is very effective in machine learning in general. Diffusion distillation is a quintessential example of this. There is still no such thing as a free lunch, but diffusion distillation enables us to cut corners with intention, and that’s worth a lot!

If you would like to cite this post in an academic context, you can use this BibTeX snippet:

@misc{dieleman2024distillation,
  author = {Dieleman, Sander},
  title = {The paradox of diffusion distillation},
  url = {https://sander.ai/2024/02/28/paradox.html},
  year = {2024}
}

Acknowledgements

Thanks once again to Bundle the bunny for modelling, and to kipply for permission to use this photograph. Thanks to Emiel Hoogeboom, Valentin De Bortoli, Pierre Richemond, Andriy Mnih and all my colleagues at Google DeepMind for various discussions, which continue to shape my thoughts on diffusion models and beyond!

References

Guidance is a powerful method that can be used to enhance diffusion model sampling. As I’ve discussed in an earlier blog post, it’s almost like a cheat code: it can improve sample quality so much that it’s as if the model had ten times the number of parameters – an order of magnitude improvement, basically for free! This follow-up post provides a geometric interpretation and visualisation of the diffusion sampling procedure, which I’ve found particularly useful to explain how guidance works.

A word of warning about high-dimensional spaces

Sampling algorithms for diffusion models typically start by initialising a canvas with random noise, and then repeatedly updating this canvas based on model predictions, until a sample from the model distribution eventually emerges.

We will represent this canvas by a vector \(\mathbf{x}_t\), where \(t\) represents the current time step in the sampling procedure. By convention, the diffusion process which gradually corrupts inputs into random noise moves forward in time from \(t=0\) to \(t=T\), so the sampling procedure goes backward in time, from \(t=T\) to \(t=0\). Therefore \(\mathbf{x}_T\) corresponds to random noise, and \(\mathbf{x}_0\) corresponds to a sample from the data distribution.

\(\mathbf{x}_t\) is a high-dimensional vector: for example, if a diffusion model produces images of size 64x64, there are 12,288 different scalar intensity values (3 colour channels per pixel). The sampling procedure then traces a path through a 12,288-dimensional Euclidean space.

It’s pretty difficult for the human brain to comprehend what that actually looks like in practice. Because our intuition is firmly rooted in our 3D surroundings, it actually tends to fail us in surprising ways in high-dimensional spaces. A while back, I wrote a blog post about some of the implications for high-dimensional probability distributions in particular. This note about why high-dimensional spheres are “spikey” is also worth a read, if you quickly want to get a feel for how weird things can get. A more thorough treatment of high-dimensional geometry can be found in chapter 2 of ‘Foundations of Data Science’¹ by Blum, Hopcroft and Kannan, which is available to download in PDF format.

Nevertheless, in this blog post, I will use diagrams that represent \(\mathbf{x}_t\) in two dimensions, because unfortunately that’s all the spatial dimensions available on your screen. This is dangerous: following our intuition in 2D might lead us to the wrong conclusions. But I’m going to do it anyway, because in spite of this, I’ve found these diagrams quite helpful to explain how manipulations such as guidance affect diffusion sampling in practice.

Here’s some advice from Geoff Hinton on dealing with high-dimensional spaces that may or may not help:

I'm laughing so hard at this slide a friend sent me from one of Geoff Hinton's courses;

"To deal with hyper-planes in a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it." pic.twitter.com/nTakZArbsD

— Robbie Barrat (@videodrome) June 10, 2018

… anyway, you’ve been warned!

Visualising diffusion sampling

To start off, let’s visualise what a step of diffusion sampling typically looks like. I will use a real photograph to which I’ve added varying amounts of noise to stand in for intermediate samples in the diffusion sampling process:

During diffusion model training, examples of noisy images are produced by taking examples of clean images from the data distribution, and adding varying amounts of noise to them. This is what I’ve done above. During sampling, we start from a canvas that is pure noise, and then the model gradually removes random noise and replaces it with meaningful structure in accordance with the data distribution. Note that I will be using this set of images to represent intermediate samples from a model, even though that’s not how they were constructed. If the model is good enough, you shouldn’t be able to tell the difference anyway!

In the diagram below, we have an intermediate noisy sample \(\mathbf{x}_t\), somewhere in the middle of the sampling process, as well as the final output of that process \(\mathbf{x}_0\), which is noise-free:

Imagine the two spatial dimensions of your screen representing just two of many thousands of pixel colour intensities (red, green or blue). Different spatial positions in the diagram correspond to different images. A single step in the sampling procedure is taken by using the model to predict where the final sample will end up. We’ll call this prediction \(\hat{\mathbf{x}}_0\):

Note how this prediction is roughly in the direction of \(\mathbf{x}_0\), but we’re not able to predict \(\mathbf{x}_0\) exactly from the current point in the sampling process, \(\mathbf{x}_t\), because the noise obscures a lot of information (especially fine-grained details), which we aren’t able to fill in all in one go. Indeed, if we were, there would be no point to this iterative sampling procedure: we could just go directly from pure noise \(\mathbf{x}_T\) to a clean image \(\mathbf{x}_0\) in one step. (As an aside, this is more or less what Consistency Models² try to achieve.)

Diffusion models estimate the expectation of \(\mathbf{x}_0\), given the current noisy input \(\mathbf{x}_t\): \(\hat{\mathbf{x}}_0 = \mathbb{E}[\mathbf{x}_0 \mid \mathbf{x}_t]\). At the highest noise levels, this expectation basically corresponds to the mean of the entire dataset, because very noisy inputs are not very informative. As a result, the prediction \(\hat{\mathbf{x}}_0\) will look like a very blurry image when visualised. At lower noise levels, this prediction will become sharper and sharper, and it will eventually resemble a sample from the data distribution. In a previous blog post, I go into a little bit more detail about why diffusion models end up estimating expectations.

In practice, diffusion models are often parameterised to predict noise, rather than clean input, which I also discussed in the same blog post. Some models also predict time-dependent linear combinations of the two. Long story short, all of these parameterisations are equivalent once the model has been trained, because a prediction of one of these quantities can be turned into a prediction of another through a linear combination of the prediction itself and the noisy input \(\mathbf{x}_t\). That’s why we can always get a prediction \(\hat{\mathbf{x}}_0\) out of any diffusion model, regardless of how it was parameterised or trained: for example, if the model predicts the noise, simply take the noisy input and subtract the predicted noise.

Diffusion model predictions also correspond to an estimate of the so-called score function, \(\nabla_{\mathbf{x}_t} \log p(\mathbf{x}_t)\). This can be interpreted as the direction in input space along which the log-likelihood of the input increases maximally. In other words, it’s the answer to the question: “how should I change the input to make it more likely?” Diffusion sampling now proceeds by taking a small step in the direction of this prediction:

This should look familiar to any machine learning practitioner, as it’s very similar to neural network training via gradient descent: backpropagation gives us the direction of steepest descent at the current point in parameter space, and at each optimisation step, we take a small step in that direction. Taking a very large step wouldn’t get us anywhere interesting, because the estimated direction is only valid locally. The same is true for diffusion sampling, except we’re now operating in the input space, rather than in the space of model parameters.

What happens next depends on the specific sampling algorithm we’ve chosen to use. There are many to choose from: DDPM³ (also called ancestral sampling), DDIM⁴, DPM++⁵ and ODE-based sampling⁶ (with many sub-variants using different ODE solvers) are just a few examples. Some of these algorithms are deterministic, which means the only source of randomness in the sampling procedure is the initial noise on the canvas. Others are stochastic, which means that further noise is injected at each step of the sampling procedure.

We’ll use DDPM as an example, because it is one of the oldest and most commonly used sampling algorithms for diffusion models. This is a stochastic algorithm, so some random noise is added after taking a step in the direction of the model prediction:

Note that I am intentionally glossing over some of the details of the sampling algorithm here (for example, the exact variance of the noise \(\varepsilon\) that is added at each step). The diagrams are schematic and the focus is on building intuition, so I think I can get away with that, but obviously it’s pretty important to get this right when you actually want to implement this algorithm.

For deterministic sampling algorithms, we can simply skip this step (i.e. set \(\varepsilon = 0\)). After this, we end up in \(\mathbf{x}_{t-1}\), which is the next iterate in the sampling procedure, and should correspond to a slightly less noisy sample. To proceed, we rinse and repeat. We can again make a prediction \(\hat{\mathbf{x}}_0\):

Because we are in a different point in input space, this prediction will also be different. Concretely, as the input to the model is now slightly less noisy, the prediction will be slightly less blurry. We now take a small step in the direction of this new prediction, and add noise to end up in \(\mathbf{x}_{t-2}\):

We can keep doing this until we eventually reach \(\mathbf{x}_0\), and we will have drawn a sample from the diffusion model. To summarise, below is an animated version of the above set of diagrams, showing the sequence of steps:

Classifier guidance

Classifier guidance^{6 7 8} provides a way to steer diffusion sampling in the direction that maximises the probability of the final sample being classified as a particular class. More broadly, this can be used to make the sample reflect any sort of conditioning signal that wasn’t provided to the diffusion model during training. In other words, it enables post-hoc conditioning.

For classifier guidance, we need an auxiliary model that predicts \(p(y \mid \mathbf{x})\), where \(y\) represents an arbitrary input feature, which could be a class label, a textual description of the input, or even a more structured object like a segmentation map or a depth map. We’ll call this model a classifier, but keep in mind that we can use many different kinds of models for this purpose, not just classifiers in the narrow sense of the word. What’s nice about this setup, is that such models are usually smaller and easier to train than diffusion models.

One important caveat is that we will be applying this auxiliary model to noisy inputs \(\mathbf{x}_t\), at varying levels of noise, so it has to be robust against this particular type of input distortion. This seems to preclude the use of off-the-shelf classifiers, and implies that we need to train a custom noise-robust classifier, or at the very least, fine-tune an off-the-shelf classifier to be noise-robust. We can also explicitly condition the classifier on the time step \(t\), so the level of noise does not have to be inferred from the input \(\mathbf{x}_t\) alone.

However, it turns out that we can construct a reasonable noise-robust classifier by combining an off-the-shelf classifier (which expects noise-free inputs) with our diffusion model. Rather than applying the classifier to \(\mathbf{x}_t\), we first predict \(\hat{\mathbf{x}}_0\) with the diffusion model, and use that as input to the classifier instead. \(\hat{\mathbf{x}}_0\) is still distorted, but by blurring rather than by Gaussian noise. Off-the-shelf classifiers tend to be much more robust to this kind of distortion out of the box. Bansal et al.⁹ named this trick “forward universal guidance”, though it has been known for some time. They also suggest some more advanced approaches for post-hoc guidance.

Using the classifier, we can now determine the direction in input space that maximises the log-likelihood of the conditioning signal, simply by computing the gradient with respect to \(\mathbf{x}_t\): \(\nabla_{\mathbf{x}_t} \log p(y \mid \mathbf{x}_t)\). (Note: if we used the above trick to construct a noise-robust classifier from an off-the-shelf one, this means we’ll need to backpropagate through the diffusion model as well.)

To apply classifier guidance, we combine the directions obtained from the diffusion model and from the classifier by adding them together, and then we take a step in this combined direction instead:

As a result, the sampling procedure will trace a different trajectory through the input space. To control the influence of the conditioning signal on the sampling procedure, we can scale the contribution of the classifier gradient by a factor \(\gamma\), which is called the guidance scale:

The combined update direction will then be influenced more strongly by the direction obtained from the classifier (provided that \(\gamma > 1\), which is usually the case):

This scale factor \(\gamma\) is an important sampling hyperparameter: if it’s too low, the effect is negligible. If it’s too high, the samples will be distorted and low-quality. This is because gradients obtained from classifiers don’t necessarily point in directions that lie on the image manifold – if we’re not careful, we may actually end up in adversarial examples, which maximise the probability of the class label but don’t actually look like an example of the class at all!

In my previous blog post on diffusion guidance, I made the connection between these operations on vectors in the input space, and the underlying manipulations of distributions they correspond to. It’s worth briefly revisiting this connection to make it more apparent:

• We’ve taken the update direction obtained from the diffusion model, which corresponds to \(\nabla_{\mathbf{x}_t} \log p_t(\mathbf{x}_t)\) (i.e. the score function), and the (scaled) update direction obtained from the classifier, \(\gamma \cdot \nabla_{\mathbf{x}_t} \log p(y \mid \mathbf{x}_t)\), and combined them simply by adding them together: \(\nabla_{\mathbf{x}_t} \log p_t(\mathbf{x}_t) + \gamma \cdot \nabla_{\mathbf{x}_t} \log p(y \mid \mathbf{x}_t)\).

• This expression corresponds to the gradient of the logarithm of \(p_t(\mathbf{x}_t) \cdot p(y \mid \mathbf{x}_t)^\gamma\).

• In other words, we have effectively reweighted the model distribution, changing the probability of each input in accordance with the probability the classifier assigns to the desired class label.

• The guidance scale \(\gamma\) corresponds to the temperature of the classifier distribution. A high temperature implies that inputs to which the classifier assigns high probabilities are upweighted more aggressively, relative to other inputs.

• The result is a new model that is much more likely to produce samples that align with the desired class label.

An animated diagram of a single step of sampling with classifier guidance is shown below:

Classifier-free guidance

Classifier-free guidance¹⁰ is a variant of guidance that does not require an auxiliary classifier model. Instead, a Bayesian classifier is constructed by combining a conditional and an unconditional generative model.

Concretely, when training a conditional generative model \(p(\mathbf{x}\mid y)\), we can drop out the conditioning \(y\) some percentage of the time (usually 10-20%) so that the same model can also act as an unconditional generative model, \(p(\mathbf{x})\). It turns out that this does not have a detrimental effect on conditional modelling performance. Using Bayes’ rule, we find that \(p(y \mid \mathbf{x}) \propto \frac{p(\mathbf{x}\mid y)}{p(\mathbf{x})}\), which gives us a way to turn our generative model into a classifier.

In diffusion models, we tend to express this in terms of score functions, rather than in terms of probability distributions. Taking the logarithm and then the gradient w.r.t. \(\mathbf{x}\), we get \(\nabla_\mathbf{x} \log p(y \mid \mathbf{x}) = \nabla_\mathbf{x} \log p(\mathbf{x} \mid y) - \nabla_\mathbf{x} \log p(\mathbf{x})\). In other words, to obtain the gradient of the classifier log-likelihood with respect to the input, all we have to do is subtract the unconditional score function from the conditional score function.

Substituting this expression into the formula for the update direction of classifier guidance, we obtain the following:

The update direction is now a linear combination of the unconditional and conditional score functions. It would be a convex combination if it were the case that \(\gamma \in [0, 1]\), but in practice \(\gamma > 1\) tends to be were the magic happens, so this is merely a barycentric combination. Note that \(\gamma = 0\) reduces to the unconditional case, and \(\gamma = 1\) reduces to the conditional (unguided) case.

How do we make sense of this geometrically? With our hybrid conditional/unconditional model, we can make two predictions \(\hat{\mathbf{x}}_0\). These will be different, because the conditioning information may allow us to make a more accurate prediction:

Next, we determine the difference vector between these two predictions. As we showed earlier, this corresponds to the gradient direction provided by the implied Bayesian classifier:

We now scale this vector by \(\gamma\):

Starting from the unconditional prediction for \(\hat{\mathbf{x}}_0\), this vector points towards a new implicit prediction, which corresponds to a stronger influence of the conditioning signal. This is the prediction we will now take a small step towards:

Classifier-free guidance tends to work a lot better than classifier guidance, because the Bayesian classifier is much more robust than a separately trained one, and the resulting update directions are much less likely to be adversarial. On top of that, it doesn’t require an auxiliary model, and generative models can be made compatible with classifier-free guidance simply through conditioning dropout during training. On the flip side, that means we can’t use this for post-hoc conditioning – all conditioning signals have to be available during training of the generative model itself. My previous blog post on guidance covers the differences in more detail.

An animated diagram of a single step of sampling with classifier-free guidance is shown below:

Closing thoughts

What’s surprising about guidance, in my opinion, is how powerful it is in practice, despite its relative simplicity. The modifications to the sampling procedure required to apply guidance are all linear operations on vectors in the input space. This is what makes it possible to interpret the procedure geometrically.

How can a set of linear operations affect the outcome of the sampling procedure so profoundly? The key is iterative refinement: these simple modifications are applied repeatedly, and crucially, they are interleaved with a very non-linear operation, which is the application of the diffusion model itself, to predict the next update direction. As a result, any linear modification of the update direction has a non-linear effect on the next update direction. Across many sampling steps, the resulting effect is highly non-linear and powerful: small differences in each step accumulate, and result in trajectories with very different endpoints.

I hope the visualisations in this post are a useful complement to my previous writing on the topic of guidance. Feel free to let me know your thoughts in the comments, or on Twitter/X (@sedielem) or Threads (@sanderdieleman).

If you would like to cite this post in an academic context, you can use this BibTeX snippet:

@misc{dieleman2023geometry,
  author = {Dieleman, Sander},
  title = {The geometry of diffusion guidance},
  url = {https://sander.ai/2023/08/28/geometry.html},
  year = {2023}
}

Acknowledgements

Thanks to Bundle for modelling and to kipply for permission to use this photograph. Thanks to my colleagues at Google DeepMind for various discussions, which continue to shape my thoughts on this topic!

References