Neurocomputing
Diffusion Probabilistic Models
1 - Denoising Diffusion probabilistic models
Generative modeling
Generative modeling consists in transforming a simple probability distribution (e.g. Gaussian) into a more complex one (e.g. images).
Learning this model allows to easily sample complex images.
VAE and GAN transform simple noise into complex distributions
Destroying information is easier than creating it
The task of the generators in GAN or VAE is very hard: going from noise to images in a few layers.
The other direction is extremely easy.
Stochastic processes can destroy information
- Iteratively adding normal noise to a signal creates a stochastic differential equation (SDE).
X_t = \sqrt{1 - p} \, X_{t-1} + \sqrt{p} \, \sigma \qquad\qquad \text{where} \qquad\qquad \sigma \sim \mathcal{N}(0, 1)
- Under some conditions, any probability distribution converges to a normal distribution.
Diffusion process
A diffusion process can iteratively destruct all information in an image through a Markov chain.
A Markov chain implies that each step is independent and governed by a probability distribution p(X_t | X_{t-1}).
Probabilistic diffusion models
- It should be possible to reverse each diffusion step by removing the noise using a form of denoising autoencoder.
Reminder: Denoising autoencoder
- A denoising autoencoder (DAE) is trained with noisy inputs but perfect desired outputs. It learns to suppress that noise.
Forward Diffusion process
The forward process iteratively corrupts the image using q(x_t | x_{t-1}) for T steps (e.g. T=1000).
The goal is to learn a reverse model p_\theta(x_{t-1} | x_t) that approximates the true q(x_{t-1} | x_t).
Forward Diffusion process
- The forward diffusion process iteratively adds Gaussian noise with a fixed schedule \beta_t:
q(x_t | x_{t-1}) = \mathcal{N}(x_t; \sqrt{1 - \beta_t} \, x_{t-1}, \beta_t I) x_t = \sqrt{1 - \beta_t} \, x_t + \beta_t \epsilon \;\; \text{where} \; \epsilon \sim \mathcal{N}(0, I)
- \mu_t = \sqrt{1 - \beta_t} \, x_t is the mean of the distribution, \sigma_t = \beta_t I its variance.
The parameter \beta_t is annealed with a decreasing schedule, as adding more noise at the end does not destroy much information.
Note that each image x_t is also a Gaussian noisy version of the original image x_0:
q(x_t | x_{0}) = \mathcal{N}(x_t; \sqrt{1 - \bar{\alpha}_t} \, x_0, \bar{\alpha}_t I) x_t = \sqrt{1 - \bar{\alpha}_t} \, x_0 + \bar{\alpha}_t \, \epsilon_t \;\; \text{where} \; \epsilon_t \sim \mathcal{N}(0, I)
with \alpha_t = 1 - \beta_t and \bar{\alpha}_t = \prod_{s=1}^t \alpha_s only depending on the history of \beta_t.
- Given the original image x_0 and a noisy version x_t, we can find the noise \epsilon_t that was added.
Probabilistic diffusion models
- The goal of the reverse diffusion process is to find a parameterized model p_\theta explaining the sequence of images backwards in time:
p_\theta(x_{0:T}) = p(x_T) \, \prod_{t=1}^T p_\theta(x_{t-1} | x_t)
where:
p_\theta(x_{t-1} | x_t) = \mathcal{N}(x_{t-1}; \mu_\theta(x_t, t), \Sigma_\theta(x_t, t))
- The reverse process is also normally distributed, given that the noise \beta_t is not too big.
Probabilistic diffusion models
- By doing some Bayesian inference on the true posterior q(x_{t-1} | x_t) = \mathcal{N}(x_{t-1}; \mu_t, \sigma_t), (Ho et al., 2020) could show that:
\begin{cases} \mu_t = \dfrac{1}{\sqrt{\alpha_t}} \, (x_t - \dfrac{1-\alpha_t}{\sqrt{1 - \bar{\alpha}_t}} \, \epsilon_t)\\ \\ \sigma_t = \dfrac{1 - \bar{\alpha}_{t-1}}{1 - \bar{\alpha}_{t}} \beta_t \, I = \bar{\beta}_t \, I \\ \end{cases}
The reverse process is also normally distributed, provided the forward noise \beta_t was not too big.
The reverse variance only depends on the schedule of \beta_t, it can be pre-computed.
Probabilistic diffusion models
- The reverse model p_\theta(x_{t-1} | x_t) only need to approximate the mean:
\mu_\theta(x_t, t) = \dfrac{1}{\sqrt{\alpha_t}} \, (x_t - \dfrac{1-\alpha_t}{\sqrt{1 - \bar{\alpha}_t}} \, \epsilon_\theta(x_t, t))
x_t is an input to the model, it does not have to predicted.
All we need to learn is the noise \epsilon_\theta(x_t, t) \approx \epsilon_t that was added to the original image x_0 to obtain x_t:
x_t = \sqrt{1 - \bar{\alpha}_t} \, x_0 + \bar{\alpha}_t \, \epsilon_t
Probabilistic diffusion models
- We want to predict the added noise in the image space:
\epsilon_\theta(x_t, t) = \epsilon_\theta(\sqrt{1 - \bar{\alpha}_t} \, x_0 + \bar{\alpha}_t \, \epsilon_t, t) \approx \epsilon_t
- We can simply minimize the mse with the true noise:
\begin{aligned} \mathcal{L}(\theta) &= \mathbb{E}_{t \sim [1, T], x_0, \epsilon_t} [(\epsilon_t - \epsilon_\theta(x_t, t))^2] \\ &= \mathbb{E}_{t \sim [1, T], x_0, \epsilon_t} [(\epsilon_t - \epsilon_\theta(\sqrt{1 - \bar{\alpha}_t} \, x_0 + \bar{\alpha}_t \, \epsilon_t, t) )^2] \\ \end{aligned}
- We only need to sample an image x_0, a time step t, a noise \epsilon_t \sim \mathcal{N}(0, I), predict the noise \epsilon_\theta(x_t, t) and minimize the mse!
- The neural network used for the reverse diffusion is usually some kind of U-net, with attentional layers, or even a vision Transformer.
Probabilistic diffusion models
- Training can be done on individual samples, no need for the whole Markov chain to create the minibatches.
Probabilistic diffusion models
- The reverse diffusion occurs iteratively backwards in time:
x_{t-1} = \dfrac{1}{\sqrt{\alpha_t}} \, (x_t - \dfrac{1-\alpha_t}{\sqrt{1 - \bar{\alpha}_t}} \, \epsilon_\theta(x_t, t)) + \sigma_t \, z
Probabilistic diffusion models
GLIDE
PDMs generate images from raw noise, but there is no control over which image will emerge.
GLIDE (Guided Language to Image Diffusion for Generation and Editing) is a PDM conditioned on a latent representation of a caption c.
As for cGAN and cVAE, the caption c is provided to the learned model:
\epsilon_\theta(x_t, t, c) \approx \epsilon_t
- Text embeddings can be obtained from any NLP model, for example a Transformer.
2 - Dall-e 2
Dall-e 2
CLIP embeddings are first learned using contrastive learning.
A conditional diffusion process (GLIDE) uses the image embeddings to produce images.
Dall-e 3, Midjourney, Stable Diffusion, etc., work on similar principles.
CLIP: Contrastive Language-Image Pre-training
Embeddings for text and images are learned using Transformer encoders and contrastive learning.
For each pair (text, image) in the training set, their representation should be made similar, while being different from the others.
Dall-e 2
- The prior network learns to map text embeddings to a sequence of image embeddings:
- After CLIP training, the two embeddings are already close from each other, but the authors find that the diffusion process works better when the image embeddings change during the diffusion.