From 7f23838b5e66276d5972cdeff79f75afd6ce2b28 Mon Sep 17 00:00:00 2001 From: NotXia <35894453+NotXia@users.noreply.github.com> Date: Sat, 14 Dec 2024 12:02:01 +0100 Subject: [PATCH] Add diffusion model proofs + conditioning --- .../img/annealed_langevin_dynamics.jpg | Bin 0 -> 32588 bytes .../img/cascaded_diffusion_models.jpg | Bin 0 -> 48367 bytes .../img/ddpm_schedule.jpg | Bin 0 -> 69618 bytes .../img/diffusion_accelerated_sampling.jpg | Bin 0 -> 25643 bytes .../img/diffusion_model_annealing.png | Bin 0 -> 140532 bytes .../diffusion_model_time_conditioning1.jpg | Bin 0 -> 74448 bytes .../diffusion_model_time_conditioning2.jpg | Bin 0 -> 75472 bytes .../img/diffusion_model_training.jpg | Bin 0 -> 69492 bytes .../img/iddpm_schedule.jpg | Bin 0 -> 67046 bytes .../img/langevin_dynamics.jpg | Bin 0 -> 43503 bytes .../img/langevin_dynamics_low_density.jpg | Bin 0 -> 381274 bytes .../img/langevin_dynamics_noise.jpg | Bin 0 -> 400391 bytes .../img/non_markovian_forward.jpg | Bin 0 -> 32527 bytes .../img/score_function.jpg | Bin 0 -> 45417 bytes .../img/unet_attention.jpg | Bin 0 -> 60689 bytes .../sections/_generative_models.tex | 609 ++++++++++++++++-- 16 files changed, 538 insertions(+), 71 deletions(-) create mode 100644 src/year2/machine-learning-for-computer-vision/img/annealed_langevin_dynamics.jpg create mode 100644 src/year2/machine-learning-for-computer-vision/img/cascaded_diffusion_models.jpg create mode 100644 src/year2/machine-learning-for-computer-vision/img/ddpm_schedule.jpg create mode 100644 src/year2/machine-learning-for-computer-vision/img/diffusion_accelerated_sampling.jpg create mode 100644 src/year2/machine-learning-for-computer-vision/img/diffusion_model_annealing.png create mode 100644 src/year2/machine-learning-for-computer-vision/img/diffusion_model_time_conditioning1.jpg create mode 100644 src/year2/machine-learning-for-computer-vision/img/diffusion_model_time_conditioning2.jpg create mode 100644 src/year2/machine-learning-for-computer-vision/img/diffusion_model_training.jpg create mode 100644 src/year2/machine-learning-for-computer-vision/img/iddpm_schedule.jpg create mode 100644 src/year2/machine-learning-for-computer-vision/img/langevin_dynamics.jpg create mode 100644 src/year2/machine-learning-for-computer-vision/img/langevin_dynamics_low_density.jpg create mode 100644 src/year2/machine-learning-for-computer-vision/img/langevin_dynamics_noise.jpg create mode 100644 src/year2/machine-learning-for-computer-vision/img/non_markovian_forward.jpg create mode 100644 src/year2/machine-learning-for-computer-vision/img/score_function.jpg create mode 100644 src/year2/machine-learning-for-computer-vision/img/unet_attention.jpg diff --git a/src/year2/machine-learning-for-computer-vision/img/annealed_langevin_dynamics.jpg b/src/year2/machine-learning-for-computer-vision/img/annealed_langevin_dynamics.jpg new file mode 100644 index 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z6D`&U1U1c=4qpV#e8N+G&j1Q##nsJ8r7BrxX$y{Z7Hlf1X^tClmhw3B~=* literal 0 HcmV?d00001 diff --git a/src/year2/machine-learning-for-computer-vision/sections/_generative_models.tex b/src/year2/machine-learning-for-computer-vision/sections/_generative_models.tex index 5e84a66..1776348 100644 --- a/src/year2/machine-learning-for-computer-vision/sections/_generative_models.tex +++ b/src/year2/machine-learning-for-computer-vision/sections/_generative_models.tex @@ -551,6 +551,11 @@ \section{Diffusion models} +\def\x{\matr{x}} +\def\params{\matr{\theta}} +\def\noise{\matr{\varepsilon}} + + \begin{description} \item[Diffusion model] \marginnote{Diffusion model} Architecture that generates an image by iteratively denoising the input latent vector. @@ -560,12 +565,12 @@ \end{remark} \item[Training] - Given an image $\matr{x}_0$, training is done in two steps: + Given an image $\x_0$, training is done in two steps: \begin{description} \item[Forward process] - The original image $\matr{x}_0$ is iteratively transformed into a latent image $\matr{x}_T$ by adding noise (i.e., transform the complex distribution $q(\matr{x}_0)$ of the original image into a simpler one $q(\matr{x}_T)$). + The original image $\x_0$ is iteratively transformed into a latent image $\x_T$ by adding noise (i.e., transform the complex distribution $q(\x_0)$ of the original image into a simpler one $q(\x_T)$). \item[Reverse process] - The latent image $\matr{x}_T$ is iteratively denoised to reconstruct the original image $\matr{x}_0$. + The latent image $\x_T$ is iteratively denoised to reconstruct the original image $\x_0$. \end{description} \begin{figure}[H] @@ -579,22 +584,22 @@ \begin{description} \item[Forward process] \marginnote{Forward process} - Given an image $\matr{x}_{t-1}$, produce a noisier version of it as: + Given an image $\x_{t-1}$, produce a noisier version of it as: \[ \begin{gathered} - \matr{x}_t = \sqrt{1-\beta_t} \matr{x}_{t-1} + \sqrt{\beta_t}\matr{\varepsilon}_t \\ - \matr{x}_t \sim q(\matr{x}_t \mid \matr{x}_{t-1}) = \mathcal{N}(\sqrt{1-\beta_t}\matr{x}_{t-1}, \beta_t\matr{I}) + \x_t = \sqrt{1-\beta_t} \x_{t-1} + \sqrt{\beta_t}\noise_t \\ + \x_t \sim q(\x_t \mid \x_{t-1}) = \mathcal{N}(\sqrt{1-\beta_t}\x_{t-1}, \beta_t\matr{I}) \end{gathered} \] where: \begin{itemize} - \item $\matr{\varepsilon}_t \sim \mathcal{N}(0; \matr{I})$ is the noise + \item $\noise_t \sim \mathcal{N}(0; \matr{I})$ is the noise \item $\beta_t \in [0,1)$ is a hyperparameter (noise schedule) and represents the variance. - \item $\sqrt{1-\beta_t} \matr{x}_{t-1}$ is the mean. + \item $\sqrt{1-\beta_t} \x_{t-1}$ is the mean. \end{itemize} \begin{remark} - $\sqrt{1-\beta_t} \matr{x}_{t-1}$ and $\beta_t$ are the mean and variance due to the fact that sampling a vector $\vec{x}$ from a Gaussian distribution with mean $\vec{\mu}$ and covariance matrix $\matr{\Sigma}$ is equivalent to: + $\sqrt{1-\beta_t} \x_{t-1}$ and $\beta_t$ are the mean and variance due to the fact that sampling a vector $\vec{x}$ from a Gaussian distribution with mean $\vec{\mu}$ and covariance matrix $\matr{\Sigma}$ is equivalent to: \[ \vec{x} = \vec{\mu} + \matr{\Sigma}^{\frac{1}{2}}\vec{y} \qquad \text{where } \vec{y} \sim \mathcal{N}(0; \matr{I}) \] If $\matr{\Sigma} = \sigma^2\matr{I}$, it holds that $\matr{\Sigma}^{\frac{1}{2}} = \sigma \matr{I}$ and we have that: \[ \vec{x} = \vec{\mu} + (\sigma\matr{I})\vec{y} \] @@ -605,20 +610,20 @@ \end{remark} \item[Diffusion kernel] \marginnote{Diffusion kernel} - It is possible to generate the latent vector $\matr{x}_t$ at time $t$ directly from $\matr{x}_0$ as: - \[ \matr{x}_t = \sqrt{\prod_{i=1}^{t}(1-\beta_i)} \cdot \matr{x}_0 + \sqrt{1-\prod_{i=1}^{t}(1-\beta_i)} \cdot \vec{\varepsilon} \qquad \text{where } \matr{\varepsilon} \sim \mathcal{N}(0; \matr{I}) \] + It is possible to generate the latent vector $\x_t$ at time $t$ directly from $\x_0$ as: + \[ \x_t = \sqrt{\prod_{i=1}^{t}(1-\beta_i)} \cdot \x_0 + \sqrt{1-\prod_{i=1}^{t}(1-\beta_i)} \cdot \noise \qquad \text{where } \noise \sim \mathcal{N}(0; \matr{I}) \] By setting the intermediate constant $\alpha_t = \prod_{i=1}^{t}(1-\beta_i)$, we have that: \[ \begin{gathered} - \matr{x}_t = \sqrt{\alpha_t} \matr{x}_0 + \sqrt{1-\alpha_t}\matr{\varepsilon} \\ - \matr{x}_t \sim q(\matr{x}_t \mid \matr{x}_0) = \mathcal{N}(\sqrt{\alpha_t}\matr{x}_0; (1-\alpha_t)\matr{I}) + \x_t = \sqrt{\alpha_t} \x_0 + \sqrt{1-\alpha_t}\noise \\ + \x_t \sim q(\x_t \mid \x_0) = \mathcal{N}(\sqrt{\alpha_t}\x_0; (1-\alpha_t)\matr{I}) \end{gathered} \] \begin{remark} As $\beta_t < 1$, it holds that $\lim\limits_{t \rightarrow +\infty} \alpha_t = 0$. In other words, for large $t = T$, only noise remains in the latent vector: - \[ q(\matr{x}_T \mid \matr{x}_0) = q(\matr{x}_T) = \mathcal{N}(0; \matr{I}) \] - Which achieves the goal of transforming a complex distribution $q(\matr{x}_0)$ into a simpler one (i.e., Gaussian). + \[ q(\x_T \mid \x_0) = q(\x_T) = \mathcal{N}(0; \matr{I}) \] + Which achieves the goal of transforming a complex distribution $q(\x_0)$ into a simpler one (i.e., Gaussian). \end{remark} \begin{example} @@ -653,90 +658,552 @@ \subsection{Reverse process} \begin{remark} - In principle, one could invert the forward process by applying Bayes rule: - \[ q(\matr{x}_{t-1} \mid \matr{x}_t) = q(\matr{x}_t \mid \matr{x}_{t-1}) \frac{q(\matr{x}_{t-1})}{q(\matr{x}_t)} \] - However, closed-form expressions for $q(\matr{x}_{t-1})$ and $q(\matr{x}_{t})$ are not available. + In principle, one could invert the forward process by applying Bayes' rule: + \[ q(\x_{t-1} \mid \x_t) = q(\x_t \mid \x_{t-1}) \frac{q(\x_{t-1})}{q(\x_t)} \] + However, closed-form expressions for $q(\x_{t-1})$ and $q(\x_{t})$ are not available. - By exploiting the Markov chain properties, it is possible to compute the conditional distribution w.r.t. $\matr{x}_0$, which is available at training time, as: + By exploiting the Markov chain properties, it is possible to compute the conditional distribution w.r.t. $\x_0$, which is available at training time, as: \[ - q(\matr{x}_{t-1} \mid \matr{x}_t, \matr{x}_0) = - q(\matr{x}_t \mid \matr{x}_{t-1}, \matr{x}_0) \frac{q(\matr{x}_{t-1} \mid \matr{x}_0)}{q(\matr{x}_t \mid \matr{x}_0)} = - \underbrace{{q(\matr{x}_t \mid \matr{x}_{t-1})}}_{\text{Forward process}} - \underbrace{\frac{q(\matr{x}_{t-1} \mid \matr{x}_0)}{q(\matr{x}_t \mid \matr{x}_0)}}_{\text{Diffusion kernels}} + q(\x_{t-1} \mid \x_t, \x_0) = + q(\x_t \mid \x_{t-1}, \x_0) \frac{q(\x_{t-1} \mid \x_0)}{q(\x_t \mid \x_0)} = + \underbrace{{q(\x_t \mid \x_{t-1})}}_{\text{Forward process}} + \underbrace{\frac{q(\x_{t-1} \mid \x_0)}{q(\x_t \mid \x_0)}}_{\text{Diffusion kernels}} \] It can be shown that this is equivalent to: - \[ q(\matr{x}_{t-1} \mid \matr{x}_t, \matr{x}_0) = \mathcal{N}\left( \frac{1-\alpha_{t-1}}{1-\alpha_t}\sqrt{1-\beta_t}\matr{x}_t + \frac{\sqrt{\alpha_{t-1}}}{1-\alpha_t}\beta_t \matr{x}_0; \frac{\beta_t(1-\alpha_{t-1})}{1-\alpha_t} \matr{I} \right) \] - However, this formulation requires knowing $\matr{x}_0$, which is only available at training time, making inference impossible. + \[ q(\x_{t-1} \mid \x_t, \x_0) = \mathcal{N}\left( \frac{1-\alpha_{t-1}}{1-\alpha_t}\sqrt{1-\beta_t}\x_t + \frac{\sqrt{\alpha_{t-1}}}{1-\alpha_t}\beta_t \x_0; \frac{\beta_t(1-\alpha_{t-1})}{1-\alpha_t} \matr{I} \right) \] + However, this formulation requires knowing $\x_0$, which is only available at training time, making inference impossible. \end{remark} \begin{description} - \item[Learned reverse process] \marginnote{Learned reverse process} - Learn a Markov chain of probabilistic mappings to reconstruct the original image $\matr{x}_0$ starting from the latent vector $\matr{x}_T$: + \item[Learned reverse process (mean)] \marginnote{Learned reverse process (mean)} + Learn a Markov chain of probabilistic mappings to reconstruct the original image $\x_0$ starting from the latent vector $\x_T$: \[ \begin{split} - p(\matr{x}_T) &= \mathcal{N}(0; \matr{I}) = q(\matr{x}_T) \\ - p(\matr{x}_{t-1} \mid \matr{x}_t) &= \mathcal{N}(\mu_t(\matr{x}_t; \matr{\theta}_t); \sigma_t\matr{I}) + p(\x_T) &= \mathcal{N}(0; \matr{I}) = q(\x_T) \\ + p(\x_{t-1} \mid \x_t) &= \mathcal{N}(\mu_t(\x_t; \params_t); \sigma_t\matr{I}) \end{split} \] where: \begin{itemize} - \item $\mu_t(\matr{x}_t; \matr{\theta}_t)$ is a neural network to estimate the mean of $p(\matr{x}_{t-1} \mid \matr{x}_t)$. + \item $\mu_t(\x_t; \params_t)$ is a neural network to estimate the mean of $p(\x_{t-1} \mid \x_t)$. \item $\sigma_t$ is, for the case of simple diffusion models, predetermined. \end{itemize} \begin{remark} - In general, $p(\matr{x}_{t-1} \mid \matr{x}_t)$ does not necessarily follow a Gaussian distribution as this is only true for $\beta_t \rightarrow 0$. However, by using small $\beta_t$ and large $T$, it can be approximately considered Gaussian. + In general, $p(\x_{t-1} \mid \x_t)$ does not necessarily follow a Gaussian distribution as this is only true for $\beta_t \rightarrow 0$. However, by using small $\beta_t$ and large $T$, it can be approximately considered Gaussian. \end{remark} \begin{description} - \item[Training] - The training objective for a set of real images $\{ \matr{x}_0^{(i)} \}_{i=1}^{I}$ is: - \[ \matr{\theta}_1^*, \dots, \matr{\theta}_T^* = \arg\max_{\matr{\theta}_1, \dots, \matr{\theta}_T} \sum_{i=1}^{I} \log\left( p(\matr{x}_0^{(i)} \mid \matr{\theta}_1, \dots, \matr{\theta}_T) \right) \] - As each image is obtained as a sequence of latents, we have that: - \[ - \begin{aligned} - p&(\matr{x}_0, \matr{x}_1, \dots, \matr{x}_T \mid \matr{\theta}_1, \dots, \matr{\theta}_T) \\ - &= p(\matr{x}_0 \mid \matr{x}_1, \dots, \matr{x}_T, \matr{\theta}_1, \dots, \matr{\theta}_T) p(\matr{x}_1, \dots, \matr{x}_T \mid \matr{\theta}_1, \dots, \matr{\theta}_T) - & p(x, y | z) = p(x | y, z)p(y | z) \\ - &= p(\matr{x}_0 \mid \matr{x}_1, \matr{\theta}_1) p(\matr{x}_1, \dots, \matr{x}_T \mid \matr{\theta}_2, \dots, \matr{\theta}_T) - & \text{Markov chain} \\ - &= \dots & \text{Repeat} \\ - &= p(\matr{x}_0 \mid \matr{x}_1, \matr{\theta}_1) \left( \prod_{t=2}^{T} p(\matr{x}_{t-1} \mid \matr{x}_t, \matr{\theta}_t) \right) p(\matr{x}_T) - \end{aligned} + \item[Loss] + The loss function for a set of images $\{ \x_0^{(i)} \}_{i=1}^{I}$ is based on the MSE of the predicted means: + \[ + \small + \begin{split} + &\mathcal{L}(\params_1, \dots, \params_T) \\ + &= \sum_{i=1}^{I}\Bigg( + -\log\left( \mathcal{N}(\x_0^{(i)}; \mu_1(\x_1^{(i)}; \params_1), \sigma_1\matr{I}) \right) + + \sum_{t=2}^{T} \frac{1}{2\sigma_t} \bigg\Vert + \matr{\mu}_{q(x_{t-1} \mid x_t, x_0)} - + \mu_t(\x_t^{(i)}; \matr{\theta_t}) \vphantom{\frac{\sqrt{0_0}}{0_0}} + \bigg\Vert^2 + \Bigg) \\ + &= \sum_{i=1}^{I}\Bigg( + \underbrace{-\log\left( \mathcal{N}(\x_0^{(i)}; \mu_1(\x_1^{(i)}; \params_1), \sigma_1\matr{I}) \right)}_{\text{Reconstruction of $x_0$ from $x_1$}} + + \sum_{t=2}^{T} \frac{1}{2\sigma_t} \bigg\Vert + \underbrace{\frac{1-\alpha_{t-1}}{1-\alpha_t} \sqrt{1-\beta_t} \x_t^{(i)} + \frac{\sqrt{\alpha_{t-1}}}{1-\alpha_t}\beta_t\x_0^{(i)}}_{\text{Ground-truth mean of $q(x_{t-1} \mid x_t, x_0)$}} - + \underbrace{\mu_t(\x_t^{(i)}; \matr{\theta_t}) \vphantom{\frac{\sqrt{0_0}}{0_0}}}_{\text{Prediction}} + \bigg\Vert^2 + \Bigg) + \end{split} \] - And the likelihood of $\matr{x}_0$ can be computed through marginalization over the latent images as follows: - \[ p(\matr{x}_0^{(i)} \mid \matr{\theta}_1, \dots, \matr{\theta}_T) = \int p(\matr{x}_0, \matr{x}_1, \dots, \matr{x}_T \mid \matr{\theta}_1, \dots, \matr{\theta}_T) \, d\matr{x}_1 \dots d\matr{x}_T \] - However, in practice this approach is computationally intractable due to the high number and high dimensionality of the latent variables. - \begin{description} - \item[Evidence lower bound (ELBO)] \marginnote{Evidence lower bound (ELBO)} - Method to compute a lower-bound of the log-likelihood. During training, we aim to maximize this bound as a proxy to maximize the likelihood. + \begin{remark} + As $T$ is usually large, the MSE term has more relevance. + \end{remark} - \begin{lemma}[Jensen's inequality] - Given a concave function $f(\cdot)$ and the expectation of the data $x$. It holds that: - \[ f(\mathbb{E}_{x \sim p(x)}[x]) \geq \mathbb{E}_{x \sim p(x)}[f(x)] \] + \begin{marginbar}{darkgray}{0}{thick} + \begin{proof} + The overall training objective for a set of real images $\{ \x_0^{(i)} \}_{i=1}^{I}$ is to maximize the likelihood of the reconstructed image: + \[ \params_1^*, \dots, \params_T^* = \arg\max_{\params_1, \dots, \params_T} \sum_{i=1}^{I} \log\left( p(\x_0^{(i)} \mid \params_1, \dots, \params_T) \right) \] - \begin{example} - Consider the logarithm function and a discrete random variable. It holds that: - \[ \log\left( \mathbb{E}_{x \sim p(x)}[x] \right) \geq \mathbb{E}_{x \sim p(x)}[\log(x)] \Rightarrow \log\left( \sum_{x \in \mathbb{X}} p(x)x \right) \geq \sum_{x \in \mathbb{X}} (p(x)\log(x)) \] - \end{example} + \indenttbox + \begin{marginbar}{darkgray}{0}{thick} + \begin{lemma}[Latents joint probabilites] \label{th:latents_joint} + As each image is obtained as a sequence of latents, we have that: + \begin{equation} + \begin{aligned} + p&(\x_0, \x_1, \dots, \x_T \mid \params_1, \dots, \params_T) \\ + &= p(\x_0 \mid \x_1, \dots, \x_T, \params_1, \dots, \params_T) p(\x_1, \dots, \x_T \mid \params_1, \dots, \params_T) + & \text{\small $p(x, y | z) = p(x | y, z)p(y | z)$} \\ + &= p(\x_0 \mid \x_1, \params_1) p(\x_1, \dots, \x_T \mid \params_2, \dots, \params_T) + & {\text{\small Markov chain}} \\ + &= \dots & {\text{\small Repeat}} \\ + &= p(\x_0 \mid \x_1, \params_1) \left( \prod_{t=2}^{T} p(\x_{t-1} \mid \x_t, \params_t) \right) p(\x_T) + \end{aligned} + \end{equation} + \end{lemma} + \end{marginbar} - \begin{figure}[H] - \centering - \includegraphics[width=0.4\linewidth]{./img/jensen_inequality.jpg} - \caption{Visualization of the Jensen's inequality} - \end{figure} - \end{lemma} + By using \Cref{th:latents_joint}, the likelihood of $\x_0$ can be computed through marginalization over the latent images as follows: + \[ p(\x_0^{(i)} \mid \params_1, \dots, \params_T) = \int p(\x_0, \x_1, \dots, \x_T \mid \params_1, \dots, \params_T) \, d\x_1 \dots d\x_T \] + However, in practice this approach is computationally intractable due to the high number and high dimensionality of the latent variables. - ELBO is computed as follows: + \indenttbox + \begin{marginbar}{darkgray}{0}{thick} + \begin{lemma}[Jensen's inequality] + Given a concave function $f(\cdot)$ and the expectation of the data $x$. It holds that: + \[ f(\mathbb{E}_{x \sim p(x)}[x]) \geq \mathbb{E}_{x \sim p(x)}[f(x)] \] + + \indenttbox + \begin{example} + Consider the logarithm function and a discrete random variable. It holds that: + \[ \log\left( \mathbb{E}_{x \sim p(x)}[x] \right) \geq \mathbb{E}_{x \sim p(x)}[\log(x)] \Rightarrow \log\left( \sum_{x \in \mathbb{X}} p(x)x \right) \geq \sum_{x \in \mathbb{X}} (p(x)\log(x)) \] + \end{example} + + \begin{figure}[H] + \centering + \includegraphics[width=0.45\linewidth]{./img/jensen_inequality.jpg} + \caption{Visualization of Jensen's inequality} + \end{figure} + \end{lemma} + \end{marginbar} + + \indenttbox + \begin{marginbar}{darkgray}{0}{thick} + \begin{lemma}[Evidence lower bound (ELBO)] \marginnote{Evidence lower bound (ELBO)} + Method to compute a lower-bound of the log-likelihood. It holds that: \[ \begin{split} - \log&(p(\matr{x}_0 \mid \matr{\theta}_1, \dots, \matr{\theta}_T)) \\ - &= \log\left( \int p(\matr{x}_0, \matr{x}_1, \dots, \matr{x}_T \mid \matr{\theta}_1, \dots, \matr{\theta}_T) \, d\matr{x}_1 \dots d\matr{x}_T \right) \\ - &= \log\left( \int \frac{q(\matr{x}_1, \dots, \matr{x}_T \mid \matr{x}_0)}{q(\matr{x}_1, \dots, \matr{x}_T \mid \matr{x}_0)} p(\matr{x}_0, \matr{x}_1, \dots, \matr{x}_T \mid \matr{\theta}_1, \dots, \matr{\theta}_T) \, d\matr{x}_1 \dots d\matr{x}_T \right) \\ - &= \log\left( \mathbb{E}_{q(\matr{x}_1, \dots, \matr{x}_T \mid \matr{x}_0)}\left[ \frac{p(\matr{x}_0, \matr{x}_1, \dots, \matr{x}_T \mid \matr{\theta}_1, \dots, \matr{\theta}_T)}{q(\matr{x}_1, \dots, \matr{x}_T \mid \matr{x}_0)} \right] \right) \\ + \log&(p(\x_0 \mid \params_1, \dots, \params_T)) \\ + &= \log\left( \int p(\x_0, \x_1, \dots, \x_T \mid \params_1, \dots, \params_T) \, d\x_1 \dots d\x_T \right) \\ + &= \log\left( \int \frac{q(\x_1, \dots, \x_T \mid \x_0)}{q(\x_1, \dots, \x_T \mid \x_0)} p(\x_0, \x_1, \dots, \x_T \mid \params_1, \dots, \params_T) \, d\x_1 \dots d\x_T \right) \\ + &= \log\left( \mathbb{E}_{q(\x_1, \dots, \x_T \mid \x_0)}\left[ \frac{p(\x_0, \x_1, \dots, \x_T \mid \params_1, \dots, \params_T)}{q(\x_1, \dots, \x_T \mid \x_0)} \right] \right) \\ \end{split} \] - \end{description} + By applying Jensen's inequality, ELBO is computed as: + \[ + \begin{split} + \log&\left( \mathbb{E}_{q(\x_1, \dots, \x_T \mid \x_0)}\left[ \frac{p(\x_0, \x_1, \dots, \x_T \mid \params_1, \dots, \params_T)}{q(\x_1, \dots, \x_T \mid \x_0)} \right] \right) \geq \\ + &\mathbb{E}_{q(\x_1, \dots, \x_T \mid \x_0)}\left[ \log\left( \frac{p(\x_0, \x_1, \dots, \x_T \mid \params_1, \dots, \params_T)}{q(\x_1, \dots, \x_T \mid \x_0)} \right) \right] = \texttt{ELBO}(\params_1, \dots, \params_T) + \end{split} + \] + \end{lemma} + \end{marginbar} + + During training, we aim to maximize ELBO as a proxy to maximize the likelihood. By applying \Cref{th:latents_joint} to the argument of the logarithm in ELBO, we have that: + \[ + \begin{aligned} + &\log\left( \frac{p(\x_0, \x_1, \dots, \x_T \mid \params_1, \dots, \params_T)}{q(\x_1, \dots, \x_T \mid \x_0)} \right) \\ + &= \log\left( \frac{p(\x_0 \mid \x_1, \params_1) \left( \prod_{t=2}^{T} p(\x_{t-1} \mid \x_t, \params_t) \right) p(\x_T)}{q(\x_1 \mid \x_0) \prod_{t=2}^T q(\x_t \mid \x_{t-1}, \x_0)} \right) \\ + &= \log\left( \frac{p(\x_0 \mid \x_1, \params_1)}{q(\x_1 \mid \x_0)} \right) + \log\left( \frac{\prod_{t=2}^{T} p(\x_{t-1} \mid \x_t, \params_t)}{\prod_{t=2}^T q(\x_t \mid \x_{t-1}, \x_0)} \right) + \log(p(\x_T)) \\ + &= \log\left( \frac{p(\x_0 \mid \x_1, \params_1)}{q(\x_1 \mid \x_0)} \right) + \log\left( \prod_{t=2}^{T} \left( \frac{p(\x_{t-1} \mid \x_t, \params_t)}{q(\x_{t-1} \mid \x_{t}, \x_0)} \frac{q(\x_{t-1} \mid \x_0)}{q(\x_t \mid \x_0)} \right) \right) + \log(p(\x_T)) + & \text{\small Bayes on denom.} \\ + \end{aligned} + \] + The second term introduced by Bayes' rule can be simplified as follows: + \[ + \begin{aligned} + \prod_{t=2}^T \frac{q(\x_{t-1} \mid \x_0)}{q(\x_t \mid \x_0)} &= \frac{q(\x_1 \mid \x_0)}{\cancel{q(\x_2 \mid \x_0)}} \frac{\cancel{q(\x_2 \mid \x_0)}}{\cancel{q(\x_3 \mid \x_0)}} \cdots \frac{\cancel{q(\x_{T-1} \mid \x_0)}}{q(\x_T \mid \x_0)} \\ + &= \frac{q(\x_1 \mid \x_0)}{q(\x_T \mid \x_0)} \\ + &= \frac{q(\x_1 \mid \x_0)}{q(\x_T)} & \text{\parbox{0.2\linewidth}{\small Time $T$ is known to be $\mathcal{N}(0; \matr{I})$}} + \end{aligned} + \] + Therefore, we have that: + \[ + \begin{aligned} + &\log\left( \frac{p(\x_0 \mid \x_1, \params_1)}{\cancel{q(\x_1 \mid \x_0)}} \right) + \log\left( \frac{\cancel{q(\x_1 \mid \x_0)}}{q(\x_T)} \prod_{t=2}^{T} \frac{p(\x_{t-1} \mid \x_t, \params_t)}{q(\x_{t-1} \mid \x_{t}, \x_0)} \right) + \log(p(\x_T)) \\ + &= \log\left( p(\x_0 \mid \x_1, \params_1) \right) + \log\left(\prod_{t=2}^{T} \frac{p(\x_{t-1} \mid \x_t, \params_t)}{q(\x_{t-1} \mid \x_{t}, \x_0)} \right) + \log\left( \frac{p(\x_T)}{q(\x_T)} \right) + & \text{\parbox{0.29\linewidth}{\small $\frac{p(\x_T)}{q(\x_T)} \approx 1$ as they are both $\mathcal{N}(0; \matr{I})$}} \\ + &= \log\left( p(\x_0 \mid \x_1, \params_1) \right) + \sum_{t=2}^{T} \log\left(\frac{p(\x_{t-1} \mid \x_t, \params_t)}{q(\x_{t-1} \mid \x_{t}, \x_0)} \right) + \end{aligned} + \] + + By going back to ELBO, we have that: + \[ + \small + \begin{aligned} + &\texttt{ELBO}(\params_1, \dots, \params_T) = \mathbb{E}_{q(\x_1, \dots, \x_T \mid \x_0)}\left[ \log\left( \frac{p(\x_0, \x_1, \dots, \x_T \mid \params_1, \dots, \params_T)}{q(\x_1, \dots, \x_T \mid \x_0)} \right) \right] \\ + &\approx \mathbb{E}_{q(\x_1, \dots, \x_T \mid \x_0)}\left[ \log\left( p(\x_0 \mid \x_1, \params_1) \right) + \sum_{t=2}^{T} \log\left(\frac{p(\x_{t-1} \mid \x_t, \params_t)}{q(\x_{t-1} \mid \x_{t}, \x_0)} \right) \right] \\ + &= \mathbb{E}_{q(\x_1 \mid \x_0)}\left[ \log\left( p(\x_0 \mid \x_1, \params_1) \right) \right] - \sum_{t=2}^{T} \mathbb{E}_{q(\x_1, \dots, \x_T \mid \x_0)}\left[ \log\left(\frac{q(\x_{t-1} \mid \x_{t}, \x_0)}{p(\x_{t-1} \mid \x_t, \params_t)} \right) \right] \\ + & & \hspace{-2.5cm}\text{\small $\mathbb{E}_{q(x, y)} = \mathbb{E}_{q(y)} \mathbb{E}_{q(x \mid y)}$} \\ + &= \mathbb{E}_{q(\x_1 \mid \x_0)}\left[ \log\left( p(\x_0 \mid \x_1, \params_1) \right) \right] - \sum_{t=2}^{T} \mathbb{E}_{q(\x_t \mid \x_0)}\mathbb{E}_{q(\x_1, \dots, \x_{t-1}, \x_{t+1}, \dots, \x_T \mid \x_t, \x_0)}\left[ \log\left(\frac{q(\x_{t-1} \mid \x_{t}, \x_0)}{p(\x_{t-1} \mid \x_t, \params_t)} \right) \right] \\ + &= \mathbb{E}_{q(\x_1 \mid \x_0)}\left[ \log\left( p(\x_0 \mid \x_1, \params_1) \right) \right] - \sum_{t=2}^{T} \mathbb{E}_{q(\x_t \mid \x_0)}\mathbb{E}_{q(\x_{t-1} \mid \x_t, \x_0)}\left[ \log\left(\frac{q(\x_{t-1} \mid \x_{t}, \x_0)}{p(\x_{t-1} \mid \x_t, \params_t)} \right) \right] \\ + &= \mathbb{E}_{q(\x_1 \mid \x_0)}\left[ \log\left( p(\x_0 \mid \x_1, \params_1) \right) \right] - \sum_{t=2}^{T} \mathbb{E}_{q(\x_t \mid \x_0)}\Big[ D_\text{KL}\big(q(\x_{t-1} \mid \x_t, \x_0) \Vert p(\x_{t-1} \mid \x_t, \params_t)\big) \Big] \\ + \end{aligned} + \] + + To make ELBO a computable loss function, we have to: + \begin{itemize} + \item Approximate expectations with Monte Carlo. + \item Expand $p$ and $q$ with their definition. + \item Expand the KL divergence. As it is between two Gaussians with constant covariance matrices, it can be computed in closed form as: + \[ + \begin{split} + D_\text{KL}&\big(q(\x_{t-1} \mid \x_t, \x_0) \Vert p(\x_{t-1} \mid \x_t, \params_t)\big) \\ + &= \frac{1}{2\sigma_t} \left\Vert \matr{\mu}_{q(\x_{t-1} \mid \x_t, \x_0)} - \matr{\mu}_{p(\x_{t-1} \mid \x_t, \params_t)} \right\Vert^2 + c \\ + &= \frac{1}{2\sigma_t} \left\Vert \left(\frac{1-\alpha_{t-1}}{1-\alpha_t} \sqrt{1-\beta_t} \x_t + \frac{\sqrt{\alpha_{t-1}}}{1-\alpha_t} \beta_t \x_0\right) - \mu_t(\x_t; \params_t) \right\Vert^2 + c + \end{split} + \] + where $c$ is a constant. + \end{itemize} + + Finally, the loss is defined from ELBO as: + \[ + \small + - \sum_{i=1}^{I}\Bigg( + \log\left( \mathcal{N}(\x_0^{(i)}; \mu_1(\x_1^{(i)}; \params_1), \sigma_1\matr{I}) \right) - + \sum_{t=2}^{T} \frac{1}{2\sigma_t} \bigg\Vert + \frac{1-\alpha_{t-1}}{1-\alpha_t} \sqrt{1-\beta_t} \x_t^{(i)} + \frac{\sqrt{\alpha_{t-1}}}{1-\alpha_t}\beta_t\x_0^{(i)} - + \mu_t(\x_t^{(i)}; \matr{\theta_t}) \vphantom{\frac{\sqrt{0_0}}{0_0}} + \bigg\Vert^2 + \Bigg) + \] + \end{proof} + \end{marginbar} \end{description} -\end{description} \ No newline at end of file + + + \item[Learned reverse process (noise)] \marginnote{Learned reverse process (noise)} + Learn a network $\varepsilon_t(\x_t; \params_t)$ to predict the noise at time $t$ instead of the mean. + + \begin{description} + \item[Loss] + The loss function is the MSE between noises: + \[ + \mathcal{L}(\params_1, \dots, \params_T) = + \sum_{i=1}^{I} \left( \sum_{t=1}^{T} \frac{\beta_t^2}{(\alpha_{t-1})(1-\beta_t)} \left\Vert \varepsilon_t\left( \sqrt{\alpha_t} \x_0^{(i)} + \sqrt{1-\alpha_t} \noise_t; \params_t \right) - \noise_t \right\Vert^2 \right) + \] + \end{description} + + \begin{remark} + In practice, this approach works better. + \end{remark} + + \begin{marginbar}{darkgray}{0}{thick} + \begin{theorem} + Predicting the noise is equivalent to predicting the mean. + + \begin{proof} + Consider the diffusion kernel: + \[ \x_t = \sqrt{\alpha_t} \x_0 + \sqrt{1-\alpha_t} \noise_t \iff \x_0 = \frac{1}{\sqrt{\alpha_t}}\x_t + \frac{\sqrt{1-\alpha_t}}{\sqrt{\alpha_t}}\noise_t \] + By substituting $\x_0$ in the definition of the mean of $q(\x_{t-1} \mid \x_t, \x_0)$, we have that: + \[ + \begin{split} + \matr{\mu}_{q(\x_{t-1} \mid \x_t, \x_0)} &= \frac{1-\alpha_{t-1}}{1-\alpha_t} \sqrt{1-\beta_t} \x_t + \frac{\sqrt{\alpha_{t-1}}}{1-\alpha_t} \beta_t \x_0 \\ + &= \frac{1-\alpha_{t-1}}{1-\alpha_t} \sqrt{1-\beta_t} \x_t + \frac{\sqrt{\alpha_{t-1}}}{1-\alpha_t} \beta_t \left( \frac{1}{\sqrt{\alpha_t}}\x_t + \frac{\sqrt{1-\alpha_t}}{\sqrt{\alpha_t}}\noise_t \right) \\ + &= \dots \\ + &= \frac{1}{\sqrt{1-\beta_t}}\x_t - \frac{\beta_t}{\sqrt{1-\alpha_{t-1}}\sqrt{1-\beta_t}} \noise_t + \end{split} + \] + Therefore, with $\varepsilon_t(\x_t; \params_t)$ it is possible to obtain the mean. + + Moreover, the MSE term of the loss becomes: + \[ + \begin{split} + &\left\Vert \matr{\mu}_{q(\x_{t-1} \mid \x_t, \x_0)} - \mu(\x_t; \params_t) \right\Vert^2 \\ + &= \left\Vert \left( \frac{1}{\sqrt{1-\beta_t}}\x_t - \frac{\beta_t}{\sqrt{1-\alpha_{t-1}}\sqrt{1-\beta_t}} \noise_t \right) - \left( \frac{1}{\sqrt{1-\beta_t}}\x_t - \frac{\beta_t}{\sqrt{1-\alpha_{t-1}}\sqrt{1-\beta_t}} \varepsilon(\x_t; \params_t) \right) \right\Vert^2 \\ + &= \frac{\beta_t^2}{\alpha_{t-1}(1-\beta_t)} \Vert \varepsilon(\x_t; \params_t) - \noise_t \Vert^2 \\ + &= \frac{\beta_t^2}{\alpha_{t-1}(1-\beta_t)} \Vert \varepsilon(\sqrt{\alpha_t}\x_0 + \sqrt{1-\alpha_t}\noise_t; \params_t) - \noise_t \Vert^2 + \end{split} + \] + + Therefore, the loss that only uses MSE computed on $I$ images is: + \[ \sum_{i=1}^{I} \left( \sum_{t=1}^{T} \frac{\beta_t^2}{(\alpha_{t-1})(1-\beta_t)} \left\Vert \varepsilon_t\left( \sqrt{\alpha_t} \x_0^{(i)} + \sqrt{1-\alpha_t} \noise_t; \params_t \right) - \noise_t \right\Vert^2 \right) \] + \end{proof} + \end{theorem} + \end{marginbar} + + \begin{figure}[H] + \centering + \includegraphics[width=0.9\linewidth]{./img/diffusion_model_training.jpg} + \caption{Diffusion models training flow} + \end{figure} +\end{description} + + +\subsection{Architecture} + +\begin{description} + \item[Generation architecture] + Standard U-Net or transformers to predict the noise. + + \begin{description} + \item[U-Net with self-attention] + Add global self-attention at the layers of the backbone where the resolution of the image is sufficiently small. It is applied as follows: + \begin{enumerate} + \item Flatten the spatial dimension to obtain $C$ 1D activations. + \item Pass the flattened activations through the self-attention layer. + \item Reshape the output to match the original activation. + \end{enumerate} + + \begin{figure}[H] + \centering + \includegraphics[width=0.7\linewidth]{./img/unet_attention.jpg} + \end{figure} + \end{description} +\end{description} + + +\begin{description} + \item[Time conditioning] \marginnote{Time conditioning} + In practice, the same network with the same set of weights is used to process each time step. Therefore, some time information has to be injected. + + Use transformer positional encoding, refined through some fully-connected layers to obtain an activation encoding time information. Then, two approaches are possible: + \begin{descriptionlist} + \item[Concatenation] + The time activation is concatenated along every spatial dimension of the image activations. + + \begin{figure}[H] + \centering + \includegraphics[width=0.75\linewidth]{./img/diffusion_model_time_conditioning1.jpg} + \end{figure} + + \item[Adaptive group normalization] + The time activation is used as the modulator for adaptive group normalization (similar mechanism to AdaIN). + + \begin{figure}[H] + \centering + \includegraphics[width=0.75\linewidth]{./img/diffusion_model_time_conditioning2.jpg} + \end{figure} + \end{descriptionlist} +\end{description} + + +\subsection{Inference} + +\begin{description} + \item[Denoising diffusion probabilistic model (DDPM)] \marginnote{Denoising diffusion probabilistic model (DDPM)} + Given a random latent $\x_T \sim \mathcal{N}(0; \matr{I})$, generation is done as follows: + \begin{enumerate} + \item For $t = T, \dots, 2$: + \begin{enumerate} + \item Compute the mean of $p(\x_{t-1} \mid \x_t)$ by predicting the noise: + \[ \matr{\mu}_t = \frac{1}{\sqrt{1-\beta_t}}\x_t - \frac{\beta_t}{\sqrt{1-\alpha_{t-1}}\sqrt{1-\beta_t}} \varepsilon_t(\x_t; \matr{\theta}) \] + \item Sample the next less noisy image from $p(\x_{t-1} \mid \x_t)$: + \[ \x_{t-1} = \matr{\mu}_t + \sigma_t \noise_t \qquad \text{with } \noise_t \sim \mathcal{N}(0; \matr{I}) \] + \end{enumerate} + \item Use the mean of $p(\x_0 \mid \x_1)$ as the output image: + \[ \x_0 = \frac{1}{\sqrt{1-\beta_t}}\x_1 - \frac{\beta_1}{\sqrt{1-\alpha_{0}}\sqrt{1-\beta_1}} \varepsilon_1(\x_1; \matr{\theta}) \] + \end{enumerate} + + \begin{remark} + In the original paper, a linear schedule for $\beta_t$ has been used. This results in a schedule for $\alpha_t$ that made the image mostly noise very quickly. + + \begin{figure}[H] + \centering + \includegraphics[width=0.9\linewidth]{./img/ddpm_schedule.jpg} + \end{figure} + \end{remark} + + \item[Improved DDPM (IDDPM)] \marginnote{Improved DDPM (IDDPM)} + Use a cosine schedule for $\alpha_t$ (with $\beta_t = 1-\frac{\alpha_t}{\alpha_{t-1}}$) so that the trajectory does not destroy the image too quickly. + \begin{figure}[H] + \centering + \includegraphics[width=0.85\linewidth]{./img/iddpm_schedule.jpg} + \end{figure} +\end{description} + +\begin{remark} + The loss of diffusion models (with DDPM) only considers the marginal $q(\x_t \mid \x_0)$: + \[ \sum_{i=1}^{I} \Bigg( \sum_{t=1}^{T} \frac{\beta_t^2}{(\alpha_{t-1})(1-\beta_t)} \Big\Vert \varepsilon_t + \big( \underbrace{\sqrt{\alpha_t} \x_0^{(i)} + \sqrt{1-\alpha_t} \noise_t; \params_t}_\text{Sampled from $q(\x_t \mid \x_0)$} \big) + - \noise_t \Big\Vert^2 \Bigg) \] + Therefore, any new family of forward processes that use this same diffusion kernel (i.e., able to sample $\x_t$ conditioned to only $\x_0$) can reuse a pre-trained DDPM model. +\end{remark} + +\begin{description} + \item[Denoising diffusion implicit model (DDIM)] \marginnote{Denoising diffusion implicit model (DDIM)} + \begin{description} + \item[Forward process] + Use a family of non-Markovian forward distributions conditioned on the real image $\x_0$ and parametrized by a positive standard deviation $\vec{\sigma}$ defined as: + \[ q_\vec{\sigma}(\x_1, \dots, \x_T \mid x_0) = q_{\sigma_T}(\x_T \mid \x_0) \prod_{t=2}^{T} q_{\sigma_t}(\x_{t-1} \mid \x_t, \x_0) \] + where: + \[ + \begin{gathered} + q_{\sigma_T}(\x_T \mid \x_0) = \mathcal{N}(0, \matr{I}) \\ + q_{\sigma_t}(\x_{t-1} \mid \x_t, \x_0) = \mathcal{N}\left( \sqrt{\alpha_{t-1}}\x_0 + \sqrt{1-\alpha_{t-1}-\alpha_t^2} \frac{\x_t - \sqrt{\alpha_t} \x_0}{\sqrt{1-\alpha_t}}; \sigma_t^2\matr{I} \right) + \end{gathered} + \] + + With this definition, it can be shown that: + \[ q_{\sigma_t}(\x_t \mid \x_0) = \mathcal{N}(\sqrt{\alpha_t}\x_0; (1-\alpha_t)\matr{I}) \] + + \begin{remark} + With a specific choice for $\vec{\sigma}$ ($\sigma_t = \sqrt{\frac{1-\alpha_{t-1}}{1-\alpha_t}}\sqrt{1-\frac{\alpha_t}{\alpha_{t-1}}}$), it is possible to obtain DDPM (i.e., DDIM is a generalization of DDPM). + + In practice, instead of tuning $\sigma_t$ directly, a proxy hyperparameter $\eta$ is used as follows: + \[ \sigma_t(\eta) = \eta \sqrt{\frac{1-\alpha_{t-1}}{1-\alpha_t}}\sqrt{1-\frac{\alpha_t}{\alpha_{t-1}}} \] + In other words, $\eta$ controls $\sigma_t$ using the DDPM model as reference (with $\eta=1$ resulting in DDPM). + \end{remark} + + \begin{remark} + With $\sigma_t \rightarrow 0$, the generation process becomes more deterministic. With $\sigma_t = 0$ ($\eta=0$), the mean is always sampled (i.e., fully deterministic). + \end{remark} + + \begin{figure}[H] + \centering + \includegraphics[width=0.9\linewidth]{./img/non_markovian_forward.jpg} + \end{figure} + + \item[Reverse process] + Given a latent $\x_t$ and a DDPM model $\varepsilon_t(\cdot; \params)$, generation at time step $t$ is done as follows: + \begin{enumerate} + \item Compute an estimate for the current time step $t$ of the real image: + \[ \hat{\x}_0 = \frac{\x_t - \sqrt{\alpha_{t-1}} \varepsilon_t(\x_t; \params)}{\sqrt{\alpha_t}} = f_\params(\x_t) \] + Note that the formula comes from the usual $\x_t = \sqrt{\alpha_t}\x_0 + \sqrt{1-\alpha_t}\noise_t$. + \item Sample the next image from: + \[ p_\params(\x_{t-1} \mid \x_t) = q_\vec{\sigma}(\x_{t-1} \mid \x_t, f_\params(\x_t)) \] + (i.e., $\x_0$ in $q_\vec{\sigma}$ has been replaced with an estimation of it). + The image is obtained as: + \[ \x_{t-1} = \matr{\mu}_{q_\vec{\sigma}} + \matr{\Sigma}_{q_\vec{\sigma}} \noise \] + \end{enumerate} + + \item[Accelerate sampling] \marginnote{Accelerate sampling} + Use a forward process that only considers a subset of time steps. This allows to skip $k$ steps in the reverse process as: + \[ + \begin{split} + p_\params(\x_{t-k} \mid \x_t) &= q_\vec{\sigma}(\x_{t-k} \mid \x_t, \x_0) \\ + &= \mathcal{N}\left( \sqrt{\alpha_{t-k}}\x_0 + \sqrt{1-\alpha_{t-k}-\sigma_t^2} \frac{\x_t-\sqrt{\alpha_t}\x_0}{\sqrt{1-\alpha_t}}; \sigma_t^2\matr{I} \right) + \end{split} + \] + + \begin{remark} + Skipped steps are actually present in the forward process as during training all steps are considered. Therefore, it is only possible to skip steps during inference. + \end{remark} + + \begin{remark} + It has been observed that determinism ($\sigma_t=0$/$\eta=0$) with accelerated generation has the best performance. + \end{remark} + + \begin{figure}[H] + \centering + \includegraphics[width=0.45\linewidth]{./img/diffusion_accelerated_sampling.jpg} + \end{figure} + \end{description} +\end{description} + + +\subsection{Interpretation of diffusion models as score estimators} + +\begin{description} + \item[Score function] \marginnote{Score function} + Given a probability density function $p(x)$, its score function is defined as: + \[ s(x) = \nabla_x\left[ \log(p(x)) \right] \] + + \begin{remark} + Differently from a probability distribution, the score function $s$ does not have to be normalized to sum $1$. Therefore, it is easier to approximate with a neural network $s(x; \theta)$. + \end{remark} + + \begin{figure}[H] + \centering + \includegraphics[width=0.65\linewidth]{./img/score_function.jpg} + \end{figure} + + The score function indicates which direction maximally increases $p(x)$. In other words, it defines a vector field that points towards the modes of $p(x)$. + + \item[Langevin dynamics] \marginnote{Langevin dynamics} + Method to sample from a score function as: + \[ \x_{t-1} = \x_t + c \nabla_\x\left[ \log(p(x)) \right] + \sqrt{2c} \noise \] + + \begin{figure}[H] + \centering + \includegraphics[width=0.3\linewidth]{./img/langevin_dynamics.jpg} + \end{figure} + + \begin{remark} + Score functions are inaccurate in low density regions. Therefore, sampling is inaccurate in areas with fewer data points. + + \begin{figure}[H] + \centering + \includegraphics[width=0.65\linewidth]{./img/langevin_dynamics_low_density.jpg} + \end{figure} + \end{remark} + + \item[Langevin dynamics with noise] \marginnote{Langevin dynamics with noise} + Add noise to the original data to make the trained score function more robust. + + \begin{figure}[H] + \centering + \includegraphics[width=0.6\linewidth]{./img/langevin_dynamics_noise.jpg} + \end{figure} + + \begin{remark} + Larger scales of noise significantly alter the original distribution. Smaller scales of noise do not cover enough low density regions. + \end{remark} + + \begin{description} + \item[Annealed Langevin dynamics] \marginnote{Annealed Langevin dynamics} + Use multiple scales of noise to estimate a family of score functions $s_t(\x_t; \params)$. Then, run a few steps of Langevin dynamics for $t=T, \dots, 1$, each time restarting from $t-1$. + + \begin{figure}[H] + \centering + \includegraphics[width=0.7\linewidth]{./img/annealed_langevin_dynamics.jpg} + \end{figure} + \end{description} + + \item[Diffusion model as score estimator] + The score function of an isotropic Gaussian distribution is: + \[ + \begin{split} + s(\x) &= \nabla_\x\left[ \log(p(\x)) \right] \qquad \text{with } \x \sim p(\x) = \mathcal{N}(\matr{\mu}; \sigma^2\matr{I}) \\ + &= \nabla_\x\left[ \log\left( \frac{1}{c} e^{-\frac{(\x-\matr{\mu})^2}{2\sigma^2}} \right) \right] \\ + &= \nabla_\x\left[ \log\left(\frac{1}{c}\right) \right] + \nabla_\x\left[ -\frac{(\x - \matr{\mu})^2}{2\sigma^2} \right] \\ + &= -\frac{\x - \matr{\mu}}{\sigma^2} + \end{split} + \] + As it holds that: + \[ \x = \matr{\mu} + \sigma\noise \iff \noise = \frac{\x - \matr{\mu}}{\sigma} \qquad \text{with } \noise \sim \mathcal{N}(0; \matr{I}) \] + The score function can be rewritten as an estimator of the Gaussian noise: + \[ s(\x) = -\frac{\x - \matr{\mu}}{\sigma^2} = -\frac{\noise}{\sigma} \] + Therefore, as diffusion models learn to predict $\varepsilon_t(\x_t; \params)$, they can be seen as a score function with a scaling factor $-\frac{1}{\sigma} = -\frac{1}{\sqrt{1-\alpha_t}}$. + + As a result, diffusion models implicitly perform annealed Langevin dynamics when generating an image. + + \begin{figure}[H] + \centering + \includegraphics[width=0.3\linewidth]{./img/diffusion_model_annealing.png} + \end{figure} +\end{description} + + +\subsection{Class conditioning} + +\begin{description} + \item[One-hot conditioning] \marginnote{One-hot conditioning} + Condition generation based on a class $c$. The model predicting noise becomes: + \[ \varepsilon_t(\x_t; c, \params) \] + Architecturally, similarly to time conditioning, the one-hot encoding of the class is refined through fully-connected layers to create an embedding that is appended to the image activations. + + \begin{remark} + This works as conditioning the likelihood with a class $c$ does not change the previous proofs. + \end{remark} + + \begin{description} + \item[Cascaded diffusion models] \marginnote{Cascaded diffusion models} + Approach to generate high resolution images starting from some class conditioning. + + Given a standard diffusion model $d_1$ and a series of super-resolution diffusion models $d_2, \dots, d_n$ with increasing resolution, the generation of an image of class $c$ is done as follows: + \begin{enumerate} + \item Use the first diffusion model $d_1$ to generate a starting low-resolution image $\matr{I}_1$ from a latent and the class $c$. + \item Iterate over the super-resolution diffusion models $i=2, \dots, n$: + \begin{enumerate} + \item Up-sample the previously generated image $\matr{I}_{i-1}$ to match the shape of the current diffusion model $d_i$. + \item Generate a higher resolution image $\matr{I}_i$ using the diffusion model $d_i$ from a latent conditioned on the class $c$ and the previous image $\matr{I}_{i-1}$ (which is concatenated along the spatial dimension of the latent). + \end{enumerate} + \end{enumerate} + + \begin{remark} + Higher-resolution models in the pipeline can be seen as detail generators. + \end{remark} + + \begin{figure}[H] + \centering + \includegraphics[width=0.9\linewidth]{./img/cascaded_diffusion_models.jpg} + \end{figure} + \end{description} +\end{description} + + + +\let\x\undefined +\let\params\undefined \ No newline at end of file