Diffusion model

Score-based generative model

Score-based generative model is another formulation of diffusion modelling. They are also called noise conditional score network (NCSN) or score-matching with Langevin dynamics (SMLD).

Score matching

The idea of score functions

Consider the problem of image generation. Let ${\displaystyle x}$ represent an image, and let ${\displaystyle q(x)}$ be the probability distribution over all possible images. If we have ${\displaystyle q(x)}$ itself, then we can say for certain how likely a certain image is. However, this is intractable in general.

Most often, we are uninterested in knowing the absolute probability of a certain image. Instead, we are usually only interested in knowing how likely a certain image is compared to its immediate neighbors — e.g. how much more likely is an image of cat compared to some small variants of it? Is it more likely if the image contains two whiskers, or three, or with some Gaussian noise added?

Consequently, we are actually quite uninterested in ${\displaystyle q(x)}$ itself, but rather, ${\displaystyle {\mathrm{\nabla }}_x\ln q(x)}$. This has two major effects:

• One, we no longer need to normalize ${\displaystyle q(x)}$, but can use any ${\displaystyle \tilde{q}(x)=Cq(x)}$, where ${\displaystyle C=\int \tilde{q}(x)dx>0}$ is any unknown constant that is of no concern to us.
• Two, we are comparing ${\displaystyle q(x)}$ neighbors ${\displaystyle q(x+dx)}$, by ${\displaystyle \frac{q(x)}{q(x+dx)}=e^{-⟨{\mathrm{\nabla }}_x\ln q,dx⟩}}$

Let the score function be ${\displaystyle s(x):={\mathrm{\nabla }}_x\ln q(x)}$; then consider what we can do with ${\displaystyle s(x)}$.

As it turns out, ${\displaystyle s(x)}$ allows us to sample from ${\displaystyle q(x)}$ using thermodynamics. Specifically, if we have a potential energy function ${\displaystyle U(x)=-\ln q(x)}$, and a lot of particles in the potential well, then the distribution at thermodynamic equilibrium is the Boltzmann distribution ${\displaystyle q_U(x)\propto e^{-U(x)/k_{B}T}=q(x{)}^{1/k_{B}T}}$. At temperature ${\displaystyle k_{B}T=1}$, the Boltzmann distribution is exactly ${\displaystyle q(x)}$.

Therefore, to model ${\displaystyle q(x)}$, we may start with a particle sampled at any convenient distribution (such as the standard Gaussian distribution), then simulate the motion of the particle forwards according to the Langevin equation $${\displaystyle d{x}_t=-{\mathrm{\nabla }}_{x_t}U(x_t)dt+d{W}_t}$$ and the Boltzmann distribution is, by Fokker-Planck equation, the unique thermodynamic equilibrium. So no matter what distribution ${\displaystyle x_0}$ has, the distribution of ${\displaystyle x_t}$ converges in distribution to ${\displaystyle q}$ as ${\displaystyle t\to \mathrm{\infty }}$.

Learning the score function

Given a density ${\displaystyle q}$, we wish to learn a score function approximation ${\displaystyle f_{\theta }\approx \mathrm{\nabla }\ln q}$. This is score matching. Typically, score matching is formalized as minimizing Fisher divergence function ${\displaystyle E_q[\Vert f_{\theta }(x)-\mathrm{\nabla }\ln q(x){\Vert }^2]}$. By expanding the integral, and performing an integration by parts, $${\displaystyle E_q[\Vert f_{\theta }(x)-\mathrm{\nabla }\ln q(x){\Vert }^2]=E_q[\Vert f_{\theta }{\Vert }^2+2\mathrm{\nabla }\cdot f_{\theta }]+C}$$giving us a loss function, also known as the Hyvärinen scoring rule, that can be minimized by stochastic gradient descent.

Annealing the score function

Suppose we need to model the distribution of images, and we want ${\displaystyle x_0\sim \mathcal{N}(0,I)}$, a white-noise image. Now, most white-noise images do not look like real images, so ${\displaystyle q(x_0)\approx 0}$ for large swaths of ${\displaystyle x_0\sim \mathcal{N}(0,I)}$. This presents a problem for learning the score function, because if there are no samples around a certain point, then we can't learn the score function at that point. If we do not know the score function ${\displaystyle {\mathrm{\nabla }}_{x_t}\ln q(x_t)}$ at that point, then we cannot impose the time-evolution equation on a particle:$${\displaystyle d{x}_t={\mathrm{\nabla }}_{x_t}\ln q(x_t)dt+d{W}_t}$$To deal with this problem, we perform annealing. If ${\displaystyle q}$ is too different from a white-noise distribution, then progressively add noise until it is indistinguishable from one. That is, we perform a forward diffusion, then learn the score function, then use the score function to perform a backward diffusion.

Continuous diffusion processes

Forward diffusion process

Consider again the forward diffusion process, but this time in continuous time:$${\displaystyle x_t=\sqrt{1-{\beta }_t}{x}_{t-1}+\sqrt{{\beta }_t}{z}_t}$$By taking the ${\displaystyle {\beta }_t\to \beta (t)dt,\sqrt{dt}{z}_t\to d{W}_t}$ limit, we obtain a continuous diffusion process, in the form of a stochastic differential equation:$${\displaystyle d{x}_t=-\frac{1}{2}\beta (t)x_{t}dt+\sqrt{\beta (t)}d{W}_t}$$where ${\displaystyle W_t}$ is a Wiener process (multidimensional Brownian motion).

Now, the equation is exactly a special case of the overdamped Langevin equation$${\displaystyle d{x}_t=-\frac{D}{k_{B}T}({\mathrm{\nabla }}_{x}U)dt+\sqrt{2D}d{W}_t}$$where ${\displaystyle D}$ is diffusion tensor, ${\displaystyle T}$ is temperature, and ${\displaystyle U}$ is potential energy field. If we substitute in ${\displaystyle D=\frac{1}{2}\beta (t)I,k_{B}T=1,U=\frac{1}{2}\Vert x{\Vert }^2}$, we recover the above equation. This explains why the phrase "Langevin dynamics" is sometimes used in diffusion models.

Now the above equation is for the stochastic motion of a single particle. Suppose we have a cloud of particles distributed according to ${\displaystyle q}$ at time ${\displaystyle t=0}$, then after a long time, the cloud of particles would settle into the stable distribution of ${\displaystyle \mathcal{N}(0,I)}$. Let ${\displaystyle {\rho }_t}$ be the density of the cloud of particles at time ${\displaystyle t}$, then we have$${\displaystyle {\rho }_0=q;{\rho }_T\approx \mathcal{N}(0,I)}$$and the goal is to somehow reverse the process, so that we can start at the end and diffuse back to the beginning.

By Fokker-Planck equation, the density of the cloud evolves according to$${\displaystyle {\mathrm{\partial }}_t\ln {\rho }_t=\frac{1}{2}\beta (t)\left(n+(x+\mathrm{\nabla }\ln {\rho }_t)\cdot \mathrm{\nabla }\ln {\rho }_t+\mathrm{\Delta }\ln {\rho }_t\right)}$$where ${\displaystyle n}$ is the dimension of space, and ${\displaystyle \mathrm{\Delta }}$ is the Laplace operator. Equivalently,$${\displaystyle {\mathrm{\partial }}_t{\rho }_t=\frac{1}{2}\beta (t)(\mathrm{\nabla }\cdot (x{\rho }_t)+\mathrm{\Delta }{\rho }_t)}$$

Backward diffusion process

If we have solved ${\displaystyle {\rho }_t}$ for time ${\displaystyle t\in [0,T]}$, then we can exactly reverse the evolution of the cloud. Suppose we start with another cloud of particles with density ${\displaystyle {\nu }_0={\rho }_T}$, and let the particles in the cloud evolve according to

$${\displaystyle d{y}_t=\frac{1}{2}\beta (T-t)y_{t}dt+\beta (T-t)\underset{\text{score function }}{\underbrace{{\mathrm{\nabla }}_{y_t}\ln {\rho }_{T-t}\left(y_t\right)}}dt+\sqrt{\beta (T-t)}d{W}_t}$$

then by plugging into the Fokker-Planck equation, we find that ${\displaystyle {\mathrm{\partial }}_t{\rho }_{T-t}={\mathrm{\partial }}_t{\nu }_t}$. Thus this cloud of points is the original cloud, evolving backwards.

Noise conditional score network (NCSN)

At the continuous limit, $${\displaystyle {\overline{\alpha }}_t=(1-{\beta }_1)\cdots (1-{\beta }_t)=e^{\sum _i\ln (1-{\beta }_i)}\to e^{-{\int }_0^t\beta (t)dt}}$$ and so $${\displaystyle x_t|x_0\sim N\left(e^{-\frac{1}{2}{\int }_0^t\beta (t)dt}{x}_0,\left(1-e^{-{\int }_0^t\beta (t)dt}\right)I\right)}$$ In particular, we see that we can directly sample from any point in the continuous diffusion process without going through the intermediate steps, by first sampling ${\displaystyle x_0\sim q,z\sim \mathcal{N}(0,I)}$, then get ${\displaystyle x_t=e^{-\frac{1}{2}{\int }_0^t\beta (t)dt}{x}_0+\left(1-e^{-{\int }_0^t\beta (t)dt}\right)z}$. That is, we can quickly sample ${\displaystyle x_t\sim {\rho }_t}$ for any ${\displaystyle t\ge 0}$.

Now, define a certain probability distribution ${\displaystyle \gamma }$ over ${\displaystyle [0,\mathrm{\infty })}$, then the score-matching loss function is defined as the expected Fisher divergence: $${\displaystyle L(\theta )=E_{t\sim \gamma ,x_t\sim {\rho }_t}[\Vert f_{\theta }(x_t,t){\Vert }^2+2\mathrm{\nabla }\cdot f_{\theta }(x_t,t)]}$$ After training, ${\displaystyle f_{\theta }(x_t,t)\approx \mathrm{\nabla }\ln {\rho }_t}$, so we can perform the backwards diffusion process by first sampling ${\displaystyle x_T\sim \mathcal{N}(0,I)}$, then integrating the SDE from ${\displaystyle t=T}$ to ${\displaystyle t=0}$: $${\displaystyle x_{t-dt}=x_t+\frac{1}{2}\beta (t)x_{t}dt+\beta (t)f_{\theta }(x_t,t)dt+\sqrt{\beta (t)}d{W}_t}$$ This may be done by any SDE integration method, such as Euler–Maruyama method.

The name "noise conditional score network" is explained thus:

• "network", because ${\displaystyle f_{\theta }}$ is implemented as a neural network.
• "score", because the output of the network is interpreted as approximating the score function ${\displaystyle \mathrm{\nabla }\ln {\rho }_t}$.
• "noise conditional", because ${\displaystyle {\rho }_t}$ is equal to ${\displaystyle {\rho }_0}$ blurred by an added Gaussian noise that increases with time, and so the score function depends on the amount of noise added.

Their equivalence

DDPM and score-based generative models are equivalent. This means that a network trained using DDPM can be used as a NCSN, and vice versa.

We know that ${\displaystyle x_t|x_0\sim N\left(\sqrt{{\overline{\alpha }}_t}{x}_0,{\sigma }_t^{2}I\right)}$, so by Tweedie's formula, we have $${\displaystyle {\mathrm{\nabla }}_{x_t}\ln q(x_t)=\frac{1}{{\sigma }_t^2}(-x_t+\sqrt{{\overline{\alpha }}_t}{E}_q[x_0|x_t])}$$ As described previously, the DDPM loss function is ${\displaystyle \sum _t{L}_{simple,t}}$ with $${\displaystyle L_{simple,t}=E_{x_0\sim q;z\sim \mathcal{N}(0,I)}\left[{\Vert {ϵ}_{\theta }(x_t,t)-z\Vert }^2\right]}$$ where ${\displaystyle x_t=\sqrt{{\overline{\alpha }}_t}{x}_0+{\sigma }_{t}z}$. By a change of variables, $${\displaystyle L_{simple,t}=E_{x_0,x_t\sim q}\left[{\Vert {ϵ}_{\theta }(x_t,t)-\frac{x_t-\sqrt{{\overline{\alpha }}_t}{x}_0}{{\sigma }_t}\Vert }^2\right]=E_{x_t\sim q,x_0\sim q(\cdot |x_t)}\left[{\Vert {ϵ}_{\theta }(x_t,t)-\frac{x_t-\sqrt{{\overline{\alpha }}_t}{x}_0}{{\sigma }_t}\Vert }^2\right]}$$ and the term inside becomes a least squares regression, so if the network actually reaches the global minimum of loss, then we have ${\displaystyle {ϵ}_{\theta }(x_t,t)=\frac{x_t-\sqrt{{\overline{\alpha }}_t}{E}_q[x_0|x_t]}{{\sigma }_t}=-{\sigma }_t{\mathrm{\nabla }}_{x_t}\ln q(x_t)}$

Thus, given a good score-based network, its predicted score is a good prediction of the noise (after scaling by ${\displaystyle {\sigma }_t}$), and thus can be used for denoising.

Conversely, the continuous limit ${\displaystyle x_{t-1}=x_{t-dt},{\beta }_t=\beta (t)dt,z_t\sqrt{dt}=d{W}_t}$ of the backward equation $${\displaystyle x_{t-1}=\frac{x_t}{\sqrt{{\alpha }_t}}-\frac{{\beta }_t}{{\sigma }_t\sqrt{{\alpha }_t}}{ϵ}_{\theta }(x_t,t)+\sqrt{{\beta }_t}{z}_t;z_t\sim \mathcal{N}(0,I)}$$ gives us precisely the same equation as score-based diffusion: $${\displaystyle x_{t-dt}=x_t(1+\beta (t)dt/2)+\beta (t){\mathrm{\nabla }}_{x_t}\ln q(x_t)dt+\sqrt{\beta (t)}d{W}_t}$$Thus, at infinitesimal steps of DDPM, a denoising network performs score-based diffusion.

Main variants

Noise schedule

In DDPM, the sequence of numbers ${\displaystyle 0={\sigma }_0<{\sigma }_1<\cdots <{\sigma }_T<1}$ is called a (discrete time) noise schedule. In general, consider a strictly increasing monotonic function ${\displaystyle \sigma }$ of type ${\displaystyle \mathbb{R}\to (0,1)}$, such as the sigmoid function. In that case, a noise schedule is a sequence of real numbers ${\displaystyle {\lambda }_1<{\lambda }_2<\cdots <{\lambda }_T}$. It then defines a sequence of noises ${\displaystyle {\sigma }_t:=\sigma ({\lambda }_t)}$, which then derives the other quantities ${\displaystyle {\beta }_t=1-\frac{1-{\sigma }_t^2}{1-{\sigma }_{t-1}^2}}$.

In order to use arbitrary noise schedules, instead of training a noise prediction model ${\displaystyle {ϵ}_{\theta }(x_t,t)}$, one trains ${\displaystyle {ϵ}_{\theta }(x_t,{\sigma }_t)}$.

Similarly, for the noise conditional score network, instead of training ${\displaystyle f_{\theta }(x_t,t)}$, one trains ${\displaystyle f_{\theta }(x_t,{\sigma }_t)}$.

Denoising Diffusion Implicit Model (DDIM)

The original DDPM method for generating images is slow, since the forward diffusion process usually takes ${\displaystyle T\sim 1000}$ to make the distribution of ${\displaystyle x_T}$ to appear close to Gaussian. However this means the backward diffusion process also take 1000 steps. Unlike the forward diffusion process, which can skip steps as ${\displaystyle x_t|x_0}$ is Gaussian for all ${\displaystyle t\ge 1}$, the backward diffusion process does not allow skipping steps. For example, to sample ${\displaystyle x_{t-2}|x_{t-1}\sim \mathcal{N}({\mu }_{\theta }(x_{t-1},t-1),{\mathrm{\Sigma }}_{\theta }(x_{t-1},t-1))}$ requires the model to first sample ${\displaystyle x_{t-1}}$. Attempting to directly sample ${\displaystyle x_{t-2}|x_t}$ would require us to marginalize out ${\displaystyle x_{t-1}}$, which is generally intractable.

DDIM is a method to take any model trained on DDPM loss, and use it to sample with some steps skipped, sacrificing an adjustable amount of quality. If we generate the Markovian chain case in DDPM to non-Markovian case, DDIM corresponds to the case that the reverse process has variance equals to 0. In other words, the reverse process (and also the forward process) is deterministic. When using fewer sampling steps, DDIM outperforms DDPM.

In detail, the DDIM sampling method is as follows. Start with the forward diffusion process ${\displaystyle x_t=\sqrt{{\overline{\alpha }}_t}{x}_0+{\sigma }_tϵ}$. Then, during the backward denoising process, given ${\displaystyle x_t,{ϵ}_{\theta }(x_t,t)}$, the original data is estimated as $${\displaystyle x_0^{\prime }=\frac{x_t-{\sigma }_t{ϵ}_{\theta }(x_t,t)}{\sqrt{{\overline{\alpha }}_t}}}$$then the backward diffusion process can jump to any step ${\displaystyle 0\le s<t}$, and the next denoised sample is $${\displaystyle x_s=\sqrt{{\overline{\alpha }}_s}{x}_0^{\prime }+\sqrt{{\sigma }_s^2-({\sigma }_s^{\prime }{)}^2}{ϵ}_{\theta }(x_t,t)+{\sigma }_s^{\prime }ϵ}$$where ${\displaystyle {\sigma }_s^{\prime }}$ is an arbitrary real number within the range ${\displaystyle [0,{\sigma }_s]}$, and ${\displaystyle ϵ\sim \mathcal{N}(0,I)}$ is a newly sampled Gaussian noise. If all ${\displaystyle {\sigma }_s^{\prime }=0}$, then the backward process becomes deterministic, and this special case of DDIM is also called "DDIM". The original paper noted that when the process is deterministic, samples generated with only 20 steps are already very similar to ones generated with 1000 steps on the high-level.

The original paper recommended defining a single "eta value" ${\displaystyle \eta \in [0,1]}$, such that ${\displaystyle {\sigma }_s^{\prime }=\eta {\tilde{\sigma }}_s}$. When ${\displaystyle \eta =1}$, this is the original DDPM. When ${\displaystyle \eta =0}$, this is the fully deterministic DDIM. For intermediate values, the process interpolates between them.

By the equivalence, the DDIM algorithm also applies for score-based diffusion models.

Latent diffusion model (LDM)

Since the diffusion model is a general method for modelling probability distributions, if one wants to model a distribution over images, one can first encode the images into a lower-dimensional space by an encoder, then use a diffusion model to model the distribution over encoded images. Then to generate an image, one can sample from the diffusion model, then use a decoder to decode it into an image.

The encoder-decoder pair is most often a variational autoencoder (VAE).

Architectural improvements

proposed various architectural improvements. For example, they proposed log-space interpolation during backward sampling. Instead of sampling from ${\displaystyle x_{t-1}\sim \mathcal{N}({\tilde{\mu }}_t(x_t,{\tilde{x}}_0),{\tilde{\sigma }}_t^{2}I)}$, they recommended sampling from ${\displaystyle \mathcal{N}({\tilde{\mu }}_t(x_t,{\tilde{x}}_0),({\sigma }_t^v{\tilde{\sigma }}_t^{1-v}{)}^{2}I)}$ for a learned parameter ${\displaystyle v}$.

In the v-prediction formalism, the noising formula ${\displaystyle x_t=\sqrt{{\overline{\alpha }}_t}{x}_0+\sqrt{1-{\overline{\alpha }}_t}{ϵ}_t}$ is reparameterised by an angle ${\displaystyle {\varphi }_t}$ such that ${\displaystyle \cos {\varphi }_t=\sqrt{{\overline{\alpha }}_t}}$ and a "velocity" defined by ${\displaystyle \cos {\varphi }_t{ϵ}_t-\sin {\varphi }_t{x}_0}$. The network is trained to predict the velocity ${\displaystyle {\hat{v}}_{\theta }}$, and denoising is by ${\displaystyle x_{{\varphi }_t-\delta }=\cos (\delta )x_{{\varphi }_t}-\sin (\delta ){\hat{v}}_{\theta }(x_{{\varphi }_t})}$. This parameterization was found to improve performance, as the model can be trained to reach total noise (i.e. ${\displaystyle {\varphi }_t={90}^{\circ }}$) and then reverse it, whereas the standard parameterization never reaches total noise since ${\displaystyle \sqrt{{\overline{\alpha }}_t}>0}$ is always true.

Classifier guidance

Classifier guidance was proposed in 2021 to improve class-conditional generation by using a classifier. The original publication used CLIP text encoders to improve text-conditional image generation.

Suppose we wish to sample not from the entire distribution of images, but conditional on the image description. We don't want to sample a generic image, but an image that fits the description "black cat with red eyes". Generally, we want to sample from the distribution ${\displaystyle p(x|y)}$, where ${\displaystyle x}$ ranges over images, and ${\displaystyle y}$ ranges over classes of images (a description "black cat with red eyes" is just a very detailed class, and a class "cat" is just a very vague description).

Taking the perspective of the noisy channel model, we can understand the process as follows: To generate an image ${\displaystyle x}$ conditional on description ${\displaystyle y}$, we imagine that the requester really had in mind an image ${\displaystyle x}$, but the image is passed through a noisy channel and came out garbled, as ${\displaystyle y}$. Image generation is then nothing but inferring which ${\displaystyle x}$ the requester had in mind.

In other words, conditional image generation is simply "translating from a textual language into a pictorial language". Then, as in noisy-channel model, we use Bayes theorem to get $${\displaystyle p(x|y)\propto p(y|x)p(x)}$$ in other words, if we have a good model of the space of all images, and a good image-to-class translator, we get a class-to-image translator "for free". In the equation for backward diffusion, the score ${\displaystyle \mathrm{\nabla }\ln p(x)}$ can be replaced by $${\displaystyle {\mathrm{\nabla }}_x\ln p(x|y)=\underset{\text{score}}{\underbrace{{\mathrm{\nabla }}_x\ln p(x)}}+\underset{\text{classifier guidance}}{\underbrace{{\mathrm{\nabla }}_x\ln p(y|x)}}}$$ where ${\displaystyle {\mathrm{\nabla }}_x\ln p(x)}$ is the score function, trained as previously described, and ${\displaystyle {\mathrm{\nabla }}_x\ln p(y|x)}$ is found by using a differentiable image classifier.

During the diffusion process, we need to condition on the time, giving$${\displaystyle {\mathrm{\nabla }}_{x_t}\ln p(x_t|y,t)={\mathrm{\nabla }}_{x_t}\ln p(y|x_t,t)+{\mathrm{\nabla }}_{x_t}\ln p(x_t|t)}$$Although, usually the classifier model does not depend on time, in which case ${\displaystyle p(y|x_t,t)=p(y|x_t)}$.

Classifier guidance is defined for the gradient of score function, thus for score-based diffusion network, but as previously noted, score-based diffusion models are equivalent to denoising models by ${\displaystyle {ϵ}_{\theta }(x_t,t)=-{\sigma }_t{\mathrm{\nabla }}_{x_t}\ln p(x_t|t)}$, and similarly, ${\displaystyle {ϵ}_{\theta }(x_t,y,t)=-{\sigma }_t{\mathrm{\nabla }}_{x_t}\ln p(x_t|y,t)}$. Therefore, classifier guidance works for denoising diffusion as well, using the modified noise prediction:$${\displaystyle {ϵ}_{\theta }(x_t,y,t)={ϵ}_{\theta }(x_t,t)-\underset{\text{classifier guidance}}{\underbrace{{\sigma }_t{\mathrm{\nabla }}_{x_t}\ln p(y|x_t,t)}}}$$

With temperature

The classifier-guided diffusion model samples from ${\displaystyle p(x|y)}$, which is concentrated around the maximum a posteriori estimate ${\displaystyle \arg \underset{x}{max}p(x|y)}$. If we want to force the model to move towards the maximum likelihood estimate ${\displaystyle \arg \underset{x}{max}p(y|x)}$, we can use $${\displaystyle p_{\gamma }(x|y)\propto p(y|x{)}^{\gamma }p(x)}$$ where ${\displaystyle \gamma >0}$ is interpretable as inverse temperature. In the context of diffusion models, it is usually called the guidance scale. A high ${\displaystyle \gamma }$ would force the model to sample from a distribution concentrated around ${\displaystyle \arg \underset{x}{max}p(y|x)}$. This sometimes improves quality of generated images.

This gives a modification to the previous equation:$${\displaystyle {\mathrm{\nabla }}_x\ln p_{\beta }(x|y)={\mathrm{\nabla }}_x\ln p(x)+\gamma {\mathrm{\nabla }}_x\ln p(y|x)}$$For denoising models, it corresponds to$${\displaystyle {ϵ}_{\theta }(x_t,y,t)={ϵ}_{\theta }(x_t,t)-\gamma {\sigma }_t{\mathrm{\nabla }}_{x_t}\ln p(y|x_t,t)}$$

Classifier-free guidance (CFG)

If we do not have a classifier ${\displaystyle p(y|x)}$, we could still extract one out of the image model itself: $${\displaystyle {\mathrm{\nabla }}_x\ln p_{\gamma }(x|y)=(1-\gamma ){\mathrm{\nabla }}_x\ln p(x)+\gamma {\mathrm{\nabla }}_x\ln p(x|y)}$$ Such a model is usually trained by presenting it with both ${\displaystyle (x,y)}$ and ${\displaystyle (x,\mathrm{N}\mathrm{o}\mathrm{n}\mathrm{e})}$, allowing it to model both ${\displaystyle {\mathrm{\nabla }}_x\ln p(x|y)}$ and ${\displaystyle {\mathrm{\nabla }}_x\ln p(x)}$.

Note that for CFG, the diffusion model cannot be merely a generative model of the entire data distribution ${\displaystyle {\mathrm{\nabla }}_x\ln p(x)}$. It must be a conditional generative model ${\displaystyle {\mathrm{\nabla }}_x\ln p(x|y)}$. For example, in stable diffusion, the diffusion backbone takes as input both a noisy model ${\displaystyle x_t}$, a time ${\displaystyle t}$, and a conditioning vector ${\displaystyle y}$ (such as a vector encoding a text prompt), and produces a noise prediction ${\displaystyle {ϵ}_{\theta }(x_t,y,t)}$.

For denoising models, it corresponds to$${\displaystyle {ϵ}_{\theta }(x_t,y,t,\gamma )={ϵ}_{\theta }(x_t,t)+\gamma ({ϵ}_{\theta }(x_t,y,t)-{ϵ}_{\theta }(x_t,t))}$$As sampled by DDIM, the algorithm can be written as$${\displaystyle \begin{array}{cc}{ϵ}_{\text{uncond}}& \leftarrow {ϵ}_{\theta }(x_t,t)\\ {ϵ}_{\text{cond}}& \leftarrow {ϵ}_{\theta }(x_t,t,c)\\ {ϵ}_{\text{CFG}}& \leftarrow {ϵ}_{\text{uncond}}+\gamma ({ϵ}_{\text{cond}}-{ϵ}_{\text{uncond}})\\ x_0& \leftarrow (x_t-{\sigma }_t{ϵ}_{\text{CFG}})/\sqrt{1-{\sigma }_t^2}\\ x_s& \leftarrow \sqrt{1-{\sigma }_s^2}{x}_0+\sqrt{{\sigma }_s^2-({\sigma }_s^{\prime }{)}^2}{ϵ}_{\text{uncond}}+{\sigma }_s^{\prime }ϵ\end{array}}$$A similar technique applies to language model sampling. Also, if the unconditional generation ${\displaystyle {ϵ}_{\text{uncond}}\leftarrow {ϵ}_{\theta }(x_t,t)}$ is replaced by ${\displaystyle {ϵ}_{\text{neg cond}}\leftarrow {ϵ}_{\theta }(x_t,t,c^{\prime })}$, then it results in negative prompting, which pushes the generation away from ${\displaystyle c^{\prime }}$ condition.

Samplers

Given a diffusion model, one may regard it either as a continuous process, and sample from it by integrating a SDE, or one can regard it as a discrete process, and sample from it by iterating the discrete steps. The choice of the "noise schedule" ${\displaystyle {\beta }_t}$ can also affect the quality of samples. A noise schedule is a function that sends a natural number to a noise level: $${\displaystyle t\mapsto {\beta }_t,t\in \{1,2,\dots \},\beta \in (0,1)}$$A noise schedule is more often specified by a map ${\displaystyle t\mapsto {\sigma }_t}$. The two definitions are equivalent, since ${\displaystyle {\beta }_t=1-\frac{1-{\sigma }_t^2}{1-{\sigma }_{t-1}^2}}$.

In the DDPM perspective, one can use the DDPM itself (with noise), or DDIM (with adjustable amount of noise). The case where one adds noise is sometimes called ancestral sampling. One can interpolate between noise and no noise. The amount of noise is denoted ${\displaystyle \eta }$ ("eta value") in the DDIM paper, with ${\displaystyle \eta =0}$ denoting no noise (as in deterministic DDIM), and ${\displaystyle \eta =1}$ denoting full noise (as in DDPM).

In the perspective of SDE, one can use any of the numerical integration methods, such as Euler–Maruyama method, Heun's method, linear multistep methods, etc. Just as in the discrete case, one can add an adjustable amount of noise during the integration.

A survey and comparison of samplers in the context of image generation is in.

Other examples

Notable variants include Poisson flow generative model, consistency model, critically damped Langevin diffusion, GenPhys, cold diffusion, etc.

Flow-based diffusion model

Abstractly speaking, the idea of diffusion model is to take an unknown probability distribution (the distribution of natural-looking images), then progressively convert it to a known probability distribution (standard Gaussian distribution), by building an absolutely continuous probability path connecting them. The probability path is in fact defined implicitly by the score function ${\displaystyle \mathrm{\nabla }\ln p_t}$.

In denoising diffusion models, the forward process adds noise, and the backward process removes noise. Both the forward and backward processes are SDEs, though the forward process is integrable in closed-form, so it can be done at no computational cost. The backward process is not integrable in closed-form, so it must be integrated step-by-step by standard SDE solvers, which can be very expensive. The probability path in diffusions model is defined through an Itô process and one can retrieve the deterministic process by using the Probability ODE flow formulation.

In flow-based diffusion models, the forward process is a deterministic flow along a time-dependent vector field, and the backward process is also a deterministic flow along the same vector field, but going backwards. Both processes are solutions to ODEs. If the vector field is well-behaved, the ODE will also be well-behaved.

Given two distributions ${\displaystyle {\pi }_0}$ and ${\displaystyle {\pi }_1}$, a flow-based model is a time-dependent velocity field ${\displaystyle v_t(x)}$ in ${\displaystyle [0,1]\times {\mathbb{R}}^d}$, such that if we start by sampling a point ${\displaystyle x\sim {\pi }_0}$, and let it move according to the velocity field: $${\displaystyle \frac{d}{dt}{\varphi }_t(x)=v_t({\varphi }_t(x))t\in [0,1],\text{starting from }{\varphi }_0(x)=x}$$ we end up with a point ${\displaystyle x_1\sim {\pi }_1}$. The solution ${\displaystyle {\varphi }_t}$ of the above ODE define a probability path ${\displaystyle p_t=[{\varphi }_t{]}_{\mathrm{\#}}{\pi }_0}$ by the pushforward measure operator. In particular, ${\displaystyle [{\varphi }_1{]}_{\mathrm{\#}}{\pi }_0={\pi }_1}$.

The probability path and the velocity field also satisfy the continuity equation, in the sense of probability distribution: $${\displaystyle {\mathrm{\partial }}_t{p}_t+\mathrm{\nabla }\cdot (v_t{p}_t)=0}$$ To construct a probability path, we start by construct a conditional probability path ${\displaystyle p_t(x|z)}$ and the corresponding conditional velocity field ${\displaystyle v_t(x|z)}$ on some conditional distribution ${\displaystyle q(z)}$. A natural choice is the Gaussian conditional probability path: $${\displaystyle p_t(x|z)=\mathcal{N}\left(m_t(z),{\zeta }_t^{2}I\right)}$$ The conditional velocity field which corresponds to the geodesic path between conditional Gaussian path is $${\displaystyle v_t(x|z)=\frac{{\zeta }_t^{\prime }}{{\zeta }_t}(x-m_t(z))+m_t^{\prime }(z)}$$ The probability path and velocity field are then computed by marginalizing

${\displaystyle p_t(x)=\int p_t(x|z)q(z)dz\text{ and }{v}_t(x)={\mathbb{E}}_{q(z)}\left[\frac{v_t(x|z)p_t(x|z)}{p_t(x)}\right]}$

Optimal transport flow

The idea of optimal transport flow is to construct a probability path minimizing the Wasserstein metric. The distribution on which we condition is an approximation of the optimal transport plan between ${\displaystyle {\pi }_0}$ and ${\displaystyle {\pi }_1}$: ${\displaystyle z=(x_0,x_1)}$ and ${\displaystyle q(z)=\mathrm{\Gamma }({\pi }_0,{\pi }_1)}$, where ${\displaystyle \mathrm{\Gamma }}$ is the optimal transport plan, which can be approximated by mini-batch optimal transport. If the batch size is not large, then the transport it computes can be very far from the true optimal transport.

Rectified flow

The idea of rectified flow is to learn a flow model such that the velocity is nearly constant along each flow path. This is beneficial, because we can integrate along such a vector field with very few steps. For example, if an ODE ${\displaystyle \dot{{\varphi }_t}(x)=v_t({\varphi }_t(x))}$ follows perfectly straight paths, it simplifies to ${\displaystyle {\varphi }_t(x)=x_0+t\cdot v_0(x_0)}$, allowing for exact solutions in one step. In practice, we cannot reach such perfection, but when the flow field is nearly so, we can take a few large steps instead of many little steps.

The general idea is to start with two distributions ${\displaystyle {\pi }_0}$ and ${\displaystyle {\pi }_1}$, then construct a flow field ${\displaystyle {\varphi }^0=\{{\varphi }_t:t\in [0,1]\}}$ from it, then repeatedly apply a "reflow" operation to obtain successive flow fields ${\displaystyle {\varphi }^1,{\varphi }^2,\dots }$, each straighter than the previous one. When the flow field is straight enough for the application, we stop.

Generally, for any time-differentiable process ${\displaystyle {\varphi }_t}$, ${\displaystyle v_t}$ can be estimated by solving: $${\displaystyle \underset{\theta }{min}{\int }_0^1{\mathbb{E}}_{x\sim p_t}\left[\Vert v_t(x,\theta )-v_t(x){\Vert }^2\right]\mathrm{d}t.}$$

In rectified flow, by injecting strong priors that intermediate trajectories are straight, it can achieve both theoretical relevance for optimal transport and computational efficiency, as ODEs with straight paths can be simulated precisely without time discretization.

Specifically, rectified flow seeks to match an ODE with the marginal distributions of the linear interpolation between points from distributions ${\displaystyle {\pi }_0}$ and ${\displaystyle {\pi }_1}$. Given observations ${\displaystyle x_0\sim {\pi }_0}$ and ${\displaystyle x_1\sim {\pi }_1}$, the canonical linear interpolation ${\displaystyle x_t=t{x}_1+(1-t)x_0,t\in [0,1]}$ yields a trivial case ${\displaystyle {\dot{x}}_t=x_1-x_0}$, which cannot be causally simulated without ${\displaystyle x_1}$. To address this, ${\displaystyle x_t}$ is "projected" into a space of causally simulatable ODEs, by minimizing the least squares loss with respect to the direction ${\displaystyle x_1-x_0}$: $${\displaystyle \underset{\theta }{min}{\int }_0^1{\mathbb{E}}_{{\pi }_0,{\pi }_1,p_t}\left[\Vert (x_1-x_0)-v_t(x_t){\Vert }^2\right]\mathrm{d}t.}$$