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mimetext/htmlrootassigneelast_run_timestampAڸئ|persist_js_state·has_pluto_hook_features$6d63c708-6f10-4a21-b57b-a9bd138b5c8crunning§runtime depends_on_skipped_cellsµpublished_object_keyserrored§cell_id$6d63c708-6f10-4a21-b57b-a9bd138b5c8cdepends_on_disabled_cells¦queued¤logsoutputbodyrScore matching
mimetext/htmlrootassigneelast_run_timestampAڸئzpersist_js_state·has_pluto_hook_features$410eac96-a384-48fa-b3f3-478be5bd236crunning§runtime ˸depends_on_skipped_cellsµpublished_object_keyserrored§cell_id$410eac96-a384-48fa-b3f3-478be5bd236cdepends_on_disabled_cells¦queued¤logsoutputbodyImprovement for conditioned diffusion for multi-modal distributions: use classifier [DN24]. Issue: need to train a classifier...
This classifier-guided strategy was replaced in [HS22] by a simpler trick: Train both a conditioned (with condition $\tau$) and an unconditioned diffusion model, and combine them with:
$$\bar{\epsilon}_t = \textcolor{purple}{\lambda} \epsilon_\theta(x_t, \sigma_t, \tau) \textcolor{purple}{+ (1 - \lambda) \epsilon_\theta(x_t, \sigma_t)} \qquad \text{with } \lambda > 1$$
Denoising with randomness
What is the order between $\sigma_{t-1}, \sigma_t$ and $\sigma_{t'}$ ?
We have $\sigma_{t-1} < \sigma_t$ by assumption. Then, because $\sigma_{t-1}$ is the geometric mean of $\sigma_{t'}$ and $\sigma_t$, it must be between so $\sigma_{t'} < \sigma_{t-1} < \sigma_t$.
What is the solution if $E$ and $D$ were linear (i.e. matrices) ?
Given a matrix of data $X$, consider the SVD $X = U\Sigma V^\top$:
$$\begin{align} \|X - DEX\|_F^2 & = \|U \Sigma V^\top - DEU \Sigma V^\top\|_F^2\\ & = \|\Sigma - U^\top DEU \Sigma\|_F^2\\ \end{align}$$
The matrix $DE$ can be any matrix of rank equal to the dimension of the latent space. So the optimal solution is $DE = U\text{Diag}(\mathbf{1}_r, \mathbf{0}_{n-r})U^\top$ where $r$ is the dimension of the latent space.
This is similar to the PCA $D\hat{E} = U\text{Diag}(\sigma_1, \ldots, \sigma_r, \mathbf{0}_{n-r})V^\top$ which is the minimizer of $\|X - D\hat{E}\|_F^2$. More precisely, given the optimal rank-$r$ solution $D = U_{:,1:r}, \hat{E} = \text{Diag}(\sigma_1, \ldots, \sigma_r)U_{:,1:r}^\top$, an optimal solution of the linear auto-encoder can be obtained with $E = \hat{E}X^\dagger$ when $X^\dagger = V\Sigma^{-1}U^\top$.
In view of this, an Auto-Encoder can be thought of as a nonlinear generalization of PCA.
[DN24] P. Dhariwal and A. Nichol. Diffusion Models Beat GANs on Image Synthesis. In: Proceedings of the 35th International Conference on Neural Information Processing Systems, NIPS '21 (Curran Associates Inc., Red Hook, NY, USA, Jun 2024); pp. 8780–8794. Accessed on Nov 26, 2024.
[HS22] J. Ho and T. Salimans. Classifier-Free Diffusion Guidance (Jul 2022), arXiv:2207.12598. Accessed on Nov 26, 2024.
We want to learn the distribution of our data represented by the random variable $X$.
The encoder maps a data point $x$ to a Gaussian distribution $Y \sim \mathcal{N}(E_\mu(x), E_{\sigma}(x))$ over the latent space
The decoder maps a latent variable $z \sim Z$ to a the Gaussian distribution $\mathcal{N}(D_\mu(z), D_\sigma(z))$
The Maximum Likelihood Estimator (MLE) maximizes the following sum over our datapoints $x$ with its ELBO:
$$\sum_x \log(f_X(x)) \ge \sum_x -D_\text{KL}((Y|X = x) \parallel Z) + \mathbb{E}[\log(f_{X|Z}(x|Y))]$$
So the MLE minimizes the loss
$$-\mathbb{E}[\log(f_{X|Z}(x|Y))].$$
with the KL-regularizer
$$D_\text{KL}((Y|X = x) \parallel Z)$$
Diffusion models are also known as energy-based models and then score-matching.
For fixed $\sigma$, the matching is $\epsilon_\theta(y) \approx \mathbb{E}[\mathcal{E}|X + \sigma \mathcal{E} = y] = -\sigma \nabla_y \log f_{X+\sigma\mathcal{E}}(y)$.
Training: sample $x$ / pick $x$ in dataset, sample $\varepsilon$, update $\theta$ to minimize (e.g., using gradient descent), the loss:
$$\mathbb{E}[\|\epsilon_\theta(X + \sigma \mathcal{E}) - \mathcal{E}\|^2]$$
Given $0 \le \mu < 1$, pick $\sigma_{t'}$ such that $\sigma_{t-1} = \sigma_t^\mu\sigma_{t'}^{1 - \mu}$ we have the following sampler of [PY24]:
$$\begin{align} x_{t - 1} & = x_t + (\textcolor{purple}{\sigma_{t'}} - \sigma_t) \epsilon_\theta(x_t, \sigma_t) \textcolor{purple}{+ \eta w_t} \qquad w_t \sim \mathcal{N}(0, I) \end{align}$$
For $\mu = 0$, $\sigma_{t'} = \sigma_{t-1}$ and we recover deterministic DDIM. For $\mu = 1/2$, we have DDPM of [HJA20]
Source: [RBL+22; Figure 3]
What should $\eta$ be ?
We want the noise level of $x_{t-1}$ to be at $\sigma_{t-1}$. So we want
$$\sigma_{t-1}^2 = \text{Var}(\sigma_{t'} \mathcal{E}_\theta(x_t, \sigma_t) + \eta W_t)$$
As $W_t$ and $\mathcal{E}_\theta(x_t, \sigma_t)$ are independent, their covariance is zero so
$$\sigma_{t-1}^2 = \sigma_{t'}^2 \text{Var}(\mathcal{E}_\theta(x_t, \sigma_t)) + \eta^2 \text{Var}(W_t)$$
Their variance is one, so we have
$$\sigma_{t-1}^2 = \sigma_{t'}^2 + \eta^2$$
This means that we should take $\eta = \sqrt{\sigma_{t-1}^2 - \sigma_{t'}^2}$.
Issue with small variance
The matching is $\epsilon_\theta(y\textcolor{red}{, \sigma}) \approx -\sigma \nabla_y \log f_{X+\sigma\mathcal{E}}(y)$.
Training: sample $x$ / pick $x$ in dataset, sample $\textcolor{red}{\sigma, }\varepsilon$, update $\theta$ to minimize (e.g., using gradient descent), the loss:
$$\mathbb{E}[\|\epsilon_\theta(X + \sigma \mathcal{E} \textcolor{red}{, \sigma}) - \mathcal{E}\|^2]$$
Animation generated with smalldiffusion using a deterministic (i.e. $w_k=0$) sampler.
Let $\sigma$ be a constant and consider a random variable $X$ with probability density function $f_X$ as well as a random Gaussian noise $\mathcal{E} \sim \mathcal{N}(0, 1)$ that is independent from $X$. If $Y = X + \sigma \mathcal{E}$ then
$$\mathbb{E}[X|Y=y] = y + \sigma^2 \nabla_y \log f_Y(y) \qquad \mathbb{E}[\mathcal{E}|Y=y] = -\sigma \nabla_y \log f_Y(y).$$
Issue with large variance
[HJA20] J. Ho, A. Jain and P. Abbeel. Denoising Diffusion Probabilistic Models. In: Proceedings of the 34th International Conference on Neural Information Processing Systems, NIPS '20 (Curran Associates Inc., Red Hook, NY, USA, Dec 2020); pp. 6840–6851. Accessed on Nov 26, 2024.
mimetext/htmlrootassigneelast_run_timestampAڸةT},persist_js_state·has_pluto_hook_features$06e194ec-42dc-46aa-8935-86c4ed04b7c8running§runtime depends_on_skipped_cellsµpublished_object_keyserrored§cell_id$06e194ec-42dc-46aa-8935-86c4ed04b7c8depends_on_disabled_cells¦queued¤logsoutputbody %
mimetext/htmlrootassigneelast_run_timestampAڸئypersist_js_state·has_pluto_hook_features$8f51a619-d0ad-42d2-bac9-acb9eab7cccbrunning§runtime))depends_on_skipped_cellsµpublished_object_keyserrored§cell_id$8f51a619-d0ad-42d2-bac9-acb9eab7cccbdepends_on_disabled_cells¦queued¤logsoutputbodyzVariational AutoEncoders (VAEs)
$\mathcal{L}(x)$ is a lower bound to $\log(f_X(x))$ as the Kullback-Leibler divergence $D_\text{KL}((Y|X = x) \parallel (Z | X = x))$ is always nonnegative.
In the context of VAEs, $Z$ represents the actual latent variable and $Y$ represents the estimated random variables, so $D_\text{KL}((Y|X = x) \parallel (Z | X = x))$ is a measure of the error made by our estimator.
Variational Auto-Encoder
We add artificial noise to improve learning:
$$\mathbb{E}[\|X - D(E_\mu(X) + \mathcal{E} \odot E_\sigma(X))\|_2^2]$$
An insightful way to model this is as follows.
Assume our dataset come from a distribution $X$ that is obtained from a latent space $Z$ using a decoder $X = D(Z)$.
If we knew the latent variable $z$, we would search for a decoder $D$ that maximizes $\mathbb{E}[\log(f_{X|Z}(x|z))]$.
Since we do not know $z$, we make an estimate of $y$ using an encoder $Y = E(X)$.
The validity of these encoder and decoder models can now be measured with a Maximum Likelihood Estimator that searches for the encoder and decoder models that maximize $\mathbb{E}[\log(f_X(X))]$.
Classifier-Free Guidance
Optical illusions
Utils
mimetext/htmlrootassigneelast_run_timestampAڸئ persist_js_state·has_pluto_hook_features$e33b16ec-4526-4dbe-affa-3e68aa508ab8running§runtime \depends_on_skipped_cellsµpublished_object_keyserrored§cell_id$e33b16ec-4526-4dbe-affa-3e68aa508ab8depends_on_disabled_cells¦queued¤logsoutputbodyIf $\sigma$ is too large then $\epsilon_\theta(y) \approx -\sigma \nabla_y \log f_{X+\sigma\mathcal{E}}(y)$ everywhere but the distribution $Y \sim X + \sigma \mathcal{E}$ is too noisy, less specific to $X$ (small signal to noise ratio). Image source.
[HJA20] J. Ho, A. Jain and P. Abbeel. Denoising Diffusion Probabilistic Models. In: Proceedings of the 34th International Conference on Neural Information Processing Systems, NIPS '20 (Curran Associates Inc., Red Hook, NY, USA, Dec 2020); pp. 6840–6851. Accessed on Nov 26, 2024.
[PY24] F. Permenter and C. Yuan. Interpreting and Improving Diffusion Models from an Optimization Perspective (Jun 2024), arXiv:2306.04848. Accessed on Nov 26, 2024.
Find encoder $E$ and decoder $D$ that minimize the loss:
$$\mathbb{E}[\|X - D(E(X))\|_2^2]$$
The code (aka latent variable) $z = E(x)$ typically has a smaller size compared to $x$ to force the model to only keep essential features.
From Tweedie's formula, $\epsilon$ is estimated to be $-\sigma \nabla_y \log f_Y(y)$.
Sampler Langevin dynamics:
$$\begin{multline} y_{k+1} = y_k + \delta_k \nabla_y \log f_Y(y_k) + \sqrt{2\delta_k} w_k\\ \text{where } w_k \sim \mathcal{N}(0, 1) \end{multline}$$
[RBL+22] R. Rombach, A. Blattmann, D. Lorenz et al. High-Resolution Image Synthesis With Latent Diffusion Models. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (2022); pp. 10684–10695. Accessed on Aug 26, 2024.
mimetext/htmlrootassigneelast_run_timestampAڸة[ persist_js_state·has_pluto_hook_features$3fde5f41-32a9-4585-9b6a-131173947346running§runtime2ڸdepends_on_skipped_cellsµpublished_object_keyserrored§cell_id$3fde5f41-32a9-4585-9b6a-131173947346depends_on_disabled_cells¦queued¤logsoutputbodymimetext/plainrootassigneelast_run_timestampAڸإ=persist_js_state·has_pluto_hook_features$4b272be9-6881-41e8-9168-3fe471527aa8running§runtimeidepends_on_skipped_cellsµpublished_object_keyserrored§cell_id$4b272be9-6881-41e8-9168-3fe471527aa8depends_on_disabled_cells¦queued¤logsoutputbodymimetext/plainrootassigneelast_run_timestampAڸاpersist_js_state·has_pluto_hook_features$44bb88d2-2fc7-4901-9fc3-890e54754932running§runtime5depends_on_skipped_cellsµpublished_object_keyserrored§cell_id$44bb88d2-2fc7-4901-9fc3-890e54754932depends_on_disabled_cells¦queued¤logsoutputbodyVTweedie's formula
[PY24] F. Permenter and C. Yuan. Interpreting and Improving Diffusion Models from an Optimization Perspective (Jun 2024), arXiv:2306.04848. Accessed on Nov 26, 2024.
mimetext/htmlrootassigneelast_run_timestampAڸةR+persist_js_state·has_pluto_hook_features$784608dc-cf8f-483b-964f-b66575dc430erunning§runtime'Qdepends_on_skipped_cellsµpublished_object_keyserrored§cell_id$784608dc-cf8f-483b-964f-b66575dc430edepends_on_disabled_cells¦queued¤logsoutputbody<Sampling
$$\begin{align} \text{Auto-Encoder} & & D(E(X))\\ \text{Variational Auto-Encoder} & & D(E(X) + \mathcal{E})\\ \text{Denoising Auto-Encoder} & & D(E(X + \sigma \mathcal{E}))\\ \end{align}$$
The goal of the diffusion model is, given a noisy image $Y = X + \sigma \mathcal{E}$ and $\sigma$, to find the noise $\mathcal{E}$. The denoising Auto-Encoder instead attempts to find the original image $X$. At the limit $\sigma \to 0$, the denoising Auto-Encoder $D(E(X + \sigma \mathcal{E}))$ is a classical Auto-Encoder, hence the name.
To train the denoising Auto-Encoder, we can use the Evidence Lower-Bound with $Y = X + \sigma \mathcal{E}$ and $Z$ such that $X = D(Z)$ [HJA20]:
$$-\log(f_X(x)) \le \mathbb{E}[-\log(f_{X|Z}(x|Y))] + D((Y|X = x) \parallel Z)$$
As $\epsilon_\theta(x_t, \sigma_t)$ is more accurate than $\epsilon_\theta(x_{t+1}, \sigma_{t+1})$, [PY24; Section 5] suggests accelerating the convergence by correcting part of the previous step. That is, we take:
$$\bar{\epsilon}_t = \textcolor{purple}{\gamma} \epsilon_\theta(x_t, \sigma_t) \textcolor{purple}{+ (1 - \gamma) \epsilon_\theta(x_{t+1}, \sigma_{t+1})}$$
with $\gamma > 1$.
Image is [PY24; Figure 3].
This can be approximated using Monte-Carlo given $L$ samples $\epsilon_1, \ldots \epsilon_L$ from the distribution $\mathcal{N}(0, I)$ as
$$\mathbb{E}[\log(f_{X|Z}(x|Y))] \approx \frac{1}{L} \sum_{i=1}^L \log(f_{X|Z}(x|E_\mu(x) + \epsilon_i \odot E_\sigma(x))).$$
In the simpler case where $D_\sigma(z) = \mathbf{1}$, we recognize the classical L2 norm:
$$\begin{align} \mathbb{E}[\log(f_{X|Z}(x|Y))] & \approx -\frac{\log(2\pi)}{2}+\frac{1}{L}\sum_{i=1}^L\|x - D_\mu(E_\mu(x) + \epsilon_i \odot E_\sigma(x))\|_2^2. \end{align}$$
Monte-Carlo sampling
Denoising Auto-Encoder
Diffusion Models
mimetext/htmlrootassigneelast_run_timestampAڸئP߰persist_js_state·has_pluto_hook_features$c64fc228-d1e3-423f-8fb3-b8e7dff9d097running§runtime depends_on_skipped_cellsµpublished_object_keyserrored§cell_id$c64fc228-d1e3-423f-8fb3-b8e7dff9d097depends_on_disabled_cells¦queued¤logsoutputbody[KW13] D. P. Kingma and M. Welling. Auto-Encoding Variational Bayes (Dec 2013). Accessed on Nov 26, 2024.
mimetext/htmlrootassigneelast_run_timestampAڸةS̰persist_js_state·has_pluto_hook_features$09829600-3488-492a-a507-53a4bbcb1cfcrunning§runtimeP~depends_on_skipped_cellsµpublished_object_keyserrored§cell_id$09829600-3488-492a-a507-53a4bbcb1cfcdepends_on_disabled_cells¦queued¤logsoutputbody
[BLL+23] R. Burgert, X. Li, A. Leite et al. Diffusion Illusions: Hiding Images in Plain Sight (Dec 2023), arXiv:2312.03817. Accessed on Nov 27, 2024.
[GPO24] D. Geng, I. Park and A. Owens. Visual Anagrams: Generating Multi-View Optical Illusions with Diffusion Models (Apr 2024), arXiv:2311.17919. Accessed on Nov 27, 2024.
Consider two deterministic functions $E_\mu, E_\sigma : \mathbb{R}^n \to \mathbb{R}^r$ and $D_\mu, D_\sigma : \mathbb{R}^r \to \mathbb{R}^n$. Suppose $X = D_\mu(Z) + \mathcal{E}_1 \odot D_\sigma(Z)$ and $Y = E_\mu(X) + \mathcal{E}_2 \odot E_\sigma(X)$ with $\mathcal{E}_1, \mathcal{E}_2 \sim \mathcal{N}(0, I)$. We have (see [KW13; Appendix B] for a proof):
$$2D_\text{KL}((Y|X = x) \parallel Z) = \|E_\mu(X)\|_2^2 + \|E_\sigma(X)\|_2^2 - r - \sum_{i=1}^r \log((E_\sigma(X))_i^2)$$
For the second part of the ELBO, we have
$$\begin{align} & \mathbb{E}[\log(f_{X|Z}(x|Y))]\\ & = \mathbb{E}[\log(f_{X|Z}(x|E_\mu(x) + \mathcal{E}_2 \odot E_\sigma(x)))]\\ & = \mathbb{E}[\log(f_{\mathcal{E}_1}(\text{Diag}(D_\sigma(E_\mu(x) + \mathcal{E}_2 \odot E_\sigma(x)))^{-1} (x - D_\mu(E_\mu(x) + \mathcal{E}_2 \odot E_\sigma(x)))))]\\ & = -\frac{\log(2\pi)}{2}+\mathbb{E}[\|\text{Diag}(D_\sigma(E_\mu(x) + \mathcal{E}_2 \odot E_\sigma(x)))^{-1} (x - D_\mu(E_\mu(x) + \mathcal{E}_2 \odot E_\sigma(x)))\|_2^2]. \end{align}$$
It was shown in [BLL+23] how to train a diffusion model to generate illusions. Suprisingly, [GPO24] showed that you don't need a specialized model and you can use a pre-trained diffusion model. Given $N$ transformations $v_1, \ldots, v_N$ [GPO24; Equation (2)]:
$$\bar{\epsilon}_t = \textcolor{purple}{\frac{1}{N} \sum_{i=1}^N v_i^{-1}(}\epsilon_\theta(\textcolor{purple}{v_i(}x_t\textcolor{purple}{)}, \sigma_t, \tau)\textcolor{purple}{)}$$
[KW13] D. P. Kingma and M. Welling. Auto-Encoding Variational Bayes (Dec 2013). Accessed on Nov 26, 2024.
mimetext/htmlrootassigneelast_run_timestampAڸةS persist_js_state·has_pluto_hook_features$ca68ba1d-5f02-4f80-8cd1-d0c297f19f58running§runtime depends_on_skipped_cellsµpublished_object_keyserrored§cell_id$ca68ba1d-5f02-4f80-8cd1-d0c297f19f58depends_on_disabled_cells¦queued¤logsoutputbodyVConditioned diffusion
Auto-Encoder
Proof
Reminder: $D_\text{KL}(A \parallel B) = \sum_{a \in \text{Dom}(A)} f_A(a) \log \frac{f_A(a)}{f_B(a)} = \mathbb{E}[\log f_A(A) - \log f_B(A)]$
$$\begin{align} D_\text{KL}((Y|X = x) \parallel (Z | X = x)) & = \mathbb{E}[\log(f_{Y|X}(Y|x)) - \log(f_{Z|X}(Y|x)) | X = x]\\ & = \mathbb{E}[\log(f_{Y|X}(Y|x)) - \log(f_{Z,X}(Y,x)) | X = x]\\ & \qquad + \log(f_{X}(x))\\ & = \mathbb{E}[\log(f_{Y|X}(Y|x)) - \log(f_{X|Z}(x|Y)) - \log(f_{Z}(Y)) | X = x]\\ & \qquad + \log(f_{X}(x))\\ & = D_\text{KL}((Y|X = x) \parallel Z) - \mathbb{E}[\log(f_{X|Z}(x|Y)) | X = x]\\ & \qquad + \log(f_{X}(x)) \end{align}$$
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@$0e01c57f-65b1-480c-a258-9a76513ac8d6metadatadisabled©show_logsîskip_as_script§cell_id$0e01c57f-65b1-480c-a258-9a76513ac8d6code_foldedäcodeHmd"""
For any random variables ``X``, ``Y`` and ``Z``, we have
```math
\log(f_X(x)) = D_\text{KL}((Y|X = x) \parallel (Z | X = x)) + \mathcal{L}(x)
```
where the *evidence lower bound* $(cite("kingma2013AutoEncoding"))
```math
\mathcal{L}(x) = -D_\text{KL}((Y|X = x) \parallel Z) + \mathbb{E}[\log(f_{X|Z}(x|Y)) | X = x]
```
"""$6fd4e448-992f-4976-82e9-6313ba038d26metadatadisabled©show_logsîskip_as_script§cell_id$6fd4e448-992f-4976-82e9-6313ba038d26code_foldedäcode md"""
Image from $(cite("permenter2024Interpretinga", "Figure 1")). Deterministic DDIM : move from a variance estimate ``\sigma_t`` to ``\sigma_{t-1}``:
```math
\begin{align}
X_t & = X_0 + \sigma_t \mathcal{E}\\
\mathbb{E}[X_{t-1}|X_t = x_{t}]
& =
\mathbb{E}[X_0|X_t = x_t] + \sigma_{t-1} \mathbb{E}[\mathcal{E}|X_t = x_t]\\
& =
\mathbb{E}[X_t|X_t = x_t] - \sigma_t \mathbb{E}[\mathcal{E}|X_t = x_t] + \sigma_{t-1} \mathbb{E}[\mathcal{E}|X_t = x_t]\\
& =
x_t + (\sigma_{t-1} - \sigma_t) \mathbb{E}[\mathcal{E}|X_t = x_t]\\
\end{align}
```
"""$a9e46466-cdbf-4539-ae36-a3113634b353metadatadisabled©show_logsîskip_as_script§cell_id$a9e46466-cdbf-4539-ae36-a3113634b353code_foldedäcodeframetitle("Acceleration")$3cdcd14a-9102-4ed4-b186-629ce183b73cmetadatadisabled©show_logsîskip_as_script§cell_id$3cdcd14a-9102-4ed4-b186-629ce183b73ccode_foldedäcodeimg("score")$abc07fcf-35f5-43eb-90aa-23ec0b581e56metadatadisabled©show_logsîskip_as_script§cell_id$abc07fcf-35f5-43eb-90aa-23ec0b581e56code_foldedäcode\qa(md"Proof",
md"""
Using the fact that ``\mathcal{E}`` is normally distributed,
we have
```math
\begin{align}
\nabla_y f_\mathcal{E}((y - x)/\sigma)
& = \frac{1}{\sqrt{2\pi}} \nabla_y \exp\left(-\frac{1}{2\sigma^2}\|x-y\|^2\right)\\
& = -\frac{\nabla_y \|x-y\|^2}{2\sigma^2} f_\mathcal{E}((y - x)/\sigma)\\
& = \frac{x-y}{\sigma^2} f_\mathcal{E}((y - x)/\sigma)\\
\end{align}
```
We have
```math
\begin{align*}
f_{X,Y}(x,y) & = f_X(x) f_\mathcal{E}((y - x)/\sigma)\\
\nabla_y f_{X,Y}(x,y) & = f_X(x) \nabla_y f_\mathcal{E}((y - x)/\sigma)\\
& = \frac{x-y}{\sigma^2} f_X(x) f_\mathcal{E}((y - x)/\sigma)\\
& = \frac{x-y}{\sigma^2} f_{X,Y}(x,y)\\
f_Y(y)
& = \int_x f_{X,Y}(x,y) \, \text{d} x\\
\nabla_y f_Y(y)
& = \int_x \nabla_y f_{X,Y}(x,y) \, \text{d} x\\
& = \int_x \frac{x-y}{\sigma^2} f_{X,Y}(x,y) \, \text{d} x\\
\sigma^2 \nabla_y f_Y(y)
& = \int_x x f_{X,Y}(x,y) \, \text{d} x - y \int_x f_{X,Y}(x,y) \, \text{d} x\\
& = \int_x x f_{X|Y}(x|y) \, \text{d} x f_{Y}(y) - y f_{Y}(y)\\
\sigma^2 \nabla_y f_Y(y) / f_{Y}(y)
& = \mathbb{E}[X | Y = y] - y\\
y + \sigma^2 \nabla_y \log f_Y(y) & = \mathbb{E}[X | Y = y].
\end{align*}
```
Now we have
```math
\begin{align}
\mathbb{E}[\mathcal{E}|Y=y]
& =
\frac{\mathbb{E}[Y|Y=y] - \mathbb{E}[X|Y=y]}{\sigma}\\
& =
\frac{
y - (y + \sigma^2 \nabla_y \log f_Y(y))
}{\sigma}\\
& =
-\sigma \nabla_y \log f_Y(y).
\end{align}
```
""")$d826a144-2c79-4c1e-ae3f-c11a4b5353d4metadatadisabled©show_logsîskip_as_script§cell_id$d826a144-2c79-4c1e-ae3f-c11a4b5353d4code_foldedäcodeimg("inaccurate.jpg")$6d63c708-6f10-4a21-b57b-a9bd138b5c8cmetadatadisabled©show_logsîskip_as_script§cell_id$6d63c708-6f10-4a21-b57b-a9bd138b5c8ccode_foldedäcode)frametitle("Evidence Lower BOund (ELBO)")$0788ac02-dab8-4ddd-9130-2a632d8d3685metadatadisabled©show_logsîskip_as_script§cell_id$0788ac02-dab8-4ddd-9130-2a632d8d3685code_foldedäcodeframetitle("Gaussian ELBO")$0886ac78-6530-417b-80a5-5410c4719dd2metadatadisabled©show_logsîskip_as_script§cell_id$0886ac78-6530-417b-80a5-5410c4719dd2code_foldedäcodemd"""
If ``\sigma`` is too small then the support of ``X + \sigma \mathcal{E}`` may not cover the whole state space → inaccurate ``\epsilon_\theta(y)`` in these parts. More formally, the loss is the Fisher divergence:
```math
\mathbb{E}[\|\epsilon_\theta(y) + \sigma \nabla_y \log f_Y(y)\|^2]
= \int_y f_Y(y) \|\epsilon_\theta(y) + \sigma \nabla_y \log f_Y(y)\|^2 \,\text{d}y
```
so it is inaccurate for ``y`` such that ``f_Y(y)`` is too small. [Image source](https://yang-song.net/blog/2021/score/).
"""$a7435857-1526-4155-b934-4c4cda9f6b30metadatadisabled©show_logsîskip_as_script§cell_id$a7435857-1526-4155-b934-4c4cda9f6b30code_foldedäcode&frametitle("Variance-dependent score")$19741723-3cc3-40df-83a3-fb2727504cd1metadatadisabled©show_logsîskip_as_script§cell_id$19741723-3cc3-40df-83a3-fb2727504cd1code_folded¤codeinclude("utils.jl")$eaf0bcaa-75d9-4d9c-9e19-ca5a8532ed35metadatadisabled©show_logsîskip_as_script§cell_id$eaf0bcaa-75d9-4d9c-9e19-ca5a8532ed35code_foldedäcodeframetitle("Score matching")$410eac96-a384-48fa-b3f3-478be5bd236cmetadatadisabled©show_logsîskip_as_script§cell_id$410eac96-a384-48fa-b3f3-478be5bd236ccode_foldedäcode@md"""
Improvement for conditioned diffusion for multi-modal distributions: use classifier $(cite("dhariwal2024Diffusion")). **Issue**: need to train a classifier...
This classifier-guided strategy was replaced in $(cite("ho2022ClassifierFree"))
by a simpler trick: Train both a conditioned (with condition ``\tau``) and an unconditioned diffusion model, and combine them with:
```math
\bar{\epsilon}_t = \textcolor{purple}{\lambda} \epsilon_\theta(x_t, \sigma_t, \tau) \textcolor{purple}{+ (1 - \lambda) \epsilon_\theta(x_t, \sigma_t)} \qquad \text{with } \lambda > 1
```
"""$c3574e26-6e50-4f08-852a-df3ae91c197fmetadatadisabled©show_logsîskip_as_script§cell_id$c3574e26-6e50-4f08-852a-df3ae91c197fcode_foldedäcode'frametitle("Denoising with randomness")$2cc866e8-fb23-4010-bec1-29d00202c7e1metadatadisabled©show_logsîskip_as_script§cell_id$2cc866e8-fb23-4010-bec1-29d00202c7e1code_foldedäcode+qa(md"What is the order between ``\sigma_{t-1}, \sigma_t`` and ``\sigma_{t'}`` ?",
md"""
We have ``\sigma_{t-1} < \sigma_t`` by assumption.
Then, because ``\sigma_{t-1}`` is the geometric mean of ``\sigma_{t'}`` and ``\sigma_t``, it must be between so
``\sigma_{t'} < \sigma_{t-1} < \sigma_t``.
""")$40f21927-1d88-41f4-b55d-6b81ca2ec211metadatadisabled©show_logsîskip_as_script§cell_id$40f21927-1d88-41f4-b55d-6b81ca2ec211code_foldedäcode qa(md"What is the solution if ``E`` and ``D`` were linear (i.e. matrices) ?",
md"""
Given a matrix of data ``X``, consider the SVD ``X = U\Sigma V^\top``:
```math
\begin{align}
\|X - DEX\|_F^2
& =
\|U \Sigma V^\top - DEU \Sigma V^\top\|_F^2\\
& =
\|\Sigma - U^\top DEU \Sigma\|_F^2\\
\end{align}
```
The matrix ``DE`` can be any matrix of rank equal to the dimension of the latent space.
So the optimal solution is ``DE = U\text{Diag}(\mathbf{1}_r, \mathbf{0}_{n-r})U^\top``
where ``r`` is the dimension of the latent space.
This is similar to the PCA ``D\hat{E} = U\text{Diag}(\sigma_1, \ldots, \sigma_r, \mathbf{0}_{n-r})V^\top`` which is the minimizer of ``\|X - D\hat{E}\|_F^2``.
More precisely, given the optimal rank-``r`` solution ``D = U_{:,1:r}, \hat{E} = \text{Diag}(\sigma_1, \ldots, \sigma_r)U_{:,1:r}^\top``,
an optimal solution of the linear auto-encoder can be obtained with ``E = \hat{E}X^\dagger`` when ``X^\dagger = V\Sigma^{-1}U^\top``.
In view of this, an Auto-Encoder can be thought of as a nonlinear generalization of PCA.
"""
)$8cf52c38-a9a6-4614-a80b-b36e9d4e563fmetadatadisabled©show_logsîskip_as_script§cell_id$8cf52c38-a9a6-4614-a80b-b36e9d4e563fcode_foldedäcode7refs(["dhariwal2024Diffusion", "ho2022ClassifierFree"])$23f3de75-0617-4232-bb71-bd9f3e355a1emetadatadisabled©show_logsîskip_as_script§cell_id$23f3de75-0617-4232-bb71-bd9f3e355a1ecode_foldedäcodemd"""
* We want to learn the distribution of our data represented by the random variable ``X``.
* The encoder maps a data point ``x`` to a Gaussian distribution ``Y \sim \mathcal{N}(E_\mu(x), E_{\sigma}(x))`` over the latent space
* The decoder maps a latent variable ``z \sim Z`` to a the Gaussian distribution ``\mathcal{N}(D_\mu(z), D_\sigma(z))``
The Maximum Likelihood Estimator (MLE) maximizes the following sum over our datapoints ``x`` with its ELBO:
```math
\sum_x \log(f_X(x)) \ge \sum_x -D_\text{KL}((Y|X = x) \parallel Z) + \mathbb{E}[\log(f_{X|Z}(x|Y))]
```
So the MLE minimizes the loss
```math
-\mathbb{E}[\log(f_{X|Z}(x|Y))].
```
with the KL-regularizer
```math
D_\text{KL}((Y|X = x) \parallel Z)
```
"""$687f5cb9-5a17-49cb-8d96-a35b66fbcba3metadatadisabled©show_logsîskip_as_script§cell_id$687f5cb9-5a17-49cb-8d96-a35b66fbcba3code_foldedäcode/md"""
Diffusion models are also known as *energy-based models* and then *score-matching*.
For **fixed** ``\sigma``, the matching is ``\epsilon_\theta(y) \approx \mathbb{E}[\mathcal{E}|X + \sigma \mathcal{E} = y] = -\sigma \nabla_y \log f_{X+\sigma\mathcal{E}}(y)``.
**Training**: sample ``x`` / pick ``x`` in dataset, sample ``\varepsilon``, update ``\theta`` to minimize (e.g., using gradient descent), the loss:
```math
\mathbb{E}[\|\epsilon_\theta(X + \sigma \mathcal{E}) - \mathcal{E}\|^2]
```
[Image source](https://yang-song.net/blog/2021/score/).
"""$45382f1f-89d7-4972-96df-1d9288a52102metadatadisabled©show_logsîskip_as_script§cell_id$45382f1f-89d7-4972-96df-1d9288a52102code_foldedäcodemd"""
Given ``0 \le \mu < 1``, pick ``\sigma_{t'}`` such that ``\sigma_{t-1} = \sigma_t^\mu\sigma_{t'}^{1 - \mu}`` we have the following sampler of $(cite(["permenter2024Interpretinga"])):
```math
\begin{align}
x_{t - 1} & = x_t + (\textcolor{purple}{\sigma_{t'}} - \sigma_t) \epsilon_\theta(x_t, \sigma_t) \textcolor{purple}{+ \eta w_t} \qquad w_t \sim \mathcal{N}(0, I)
\end{align}
```
For ``\mu = 0``, ``\sigma_{t'} = \sigma_{t-1}`` and we recover deterministic DDIM.
For ``\mu = 1/2``, we have DDPM of $(cite("ho2020Denoising"))
"""$e0ba3b61-a618-49ce-9466-e6a8f674bd53metadatadisabled©show_logsîskip_as_script§cell_id$e0ba3b61-a618-49ce-9466-e6a8f674bd53code_foldedäcode@md"""Source: $(cite("rombach2022HighResolution", "Figure 3"))"""$e50cffb2-ac51-483a-92d0-a19531ebb547metadatadisabled©show_logsîskip_as_script§cell_id$e50cffb2-ac51-483a-92d0-a19531ebb547code_foldedäcodeVqa(md"What should ``\eta`` be ?",
md"""
We want the noise level of ``x_{t-1}`` to be at ``\sigma_{t-1}``.
So we want
```math
\sigma_{t-1}^2 = \text{Var}(\sigma_{t'} \mathcal{E}_\theta(x_t, \sigma_t) + \eta W_t)
```
As ``W_t`` and ``\mathcal{E}_\theta(x_t, \sigma_t)`` are independent, their covariance is zero so
```math
\sigma_{t-1}^2 = \sigma_{t'}^2 \text{Var}(\mathcal{E}_\theta(x_t, \sigma_t)) + \eta^2 \text{Var}(W_t)
```
Their variance is one, so we have
```math
\sigma_{t-1}^2 = \sigma_{t'}^2 + \eta^2
```
This means that we should take ``\eta = \sqrt{\sigma_{t-1}^2 - \sigma_{t'}^2}``.
""")$a835f101-bfd0-428e-adba-64f5de7af14cmetadatadisabled©show_logsîskip_as_script§cell_id$a835f101-bfd0-428e-adba-64f5de7af14ccode_foldedäcode'frametitle("Issue with small variance")$d077a471-6b43-41df-8218-e3b4a7e38550metadatadisabled©show_logsîskip_as_script§cell_id$d077a471-6b43-41df-8218-e3b4a7e38550code_folded¤code(cite(args...) = bibcite(biblio, args...)$c3af23f7-8919-4e27-a97e-6013038b8d8cmetadatadisabled©show_logsîskip_as_script§cell_id$c3af23f7-8919-4e27-a97e-6013038b8d8ccode_foldedäcodeIHAlign(md"""
The matching is ``\epsilon_\theta(y\textcolor{red}{, \sigma}) \approx -\sigma \nabla_y \log f_{X+\sigma\mathcal{E}}(y)``.
**Training**: sample ``x`` / pick ``x`` in dataset, sample ``\textcolor{red}{\sigma, }\varepsilon``, update ``\theta`` to minimize (e.g., using gradient descent), the loss:
```math
\mathbb{E}[\|\epsilon_\theta(X + \sigma \mathcal{E} \textcolor{red}{, \sigma}) - \mathcal{E}\|^2]
```
Animation generated with [`smalldiffusion`](https://github.com/yuanchenyang/smalldiffusion) using a deterministic (i.e. ``w_k=0``) sampler.
""",
img("spiral.gif"),
)$423874b8-6d4c-423a-bf45-39d590600066metadatadisabled©show_logsîskip_as_script§cell_id$423874b8-6d4c-423a-bf45-39d590600066code_foldedäcodemd"""
Let ``\sigma`` be a constant and consider a random variable ``X`` with probability density function ``f_X`` as well as a random Gaussian noise ``\mathcal{E} \sim \mathcal{N}(0, 1)`` that is independent from ``X``.
If ``Y = X + \sigma \mathcal{E}`` then
```math
\mathbb{E}[X|Y=y] = y + \sigma^2 \nabla_y \log f_Y(y) \qquad
\mathbb{E}[\mathcal{E}|Y=y] = -\sigma \nabla_y \log f_Y(y).
```
"""$7e7e2845-4bee-4dbc-b0f9-d964cd362248metadatadisabled©show_logsîskip_as_script§cell_id$7e7e2845-4bee-4dbc-b0f9-d964cd362248code_foldedäcode'frametitle("Issue with large variance")$53b3d56e-f11d-44c2-96ca-3714f4b4f929metadatadisabled©show_logsîskip_as_script§cell_id$53b3d56e-f11d-44c2-96ca-3714f4b4f929code_foldedäcoderefs("ho2020Denoising")$06e194ec-42dc-46aa-8935-86c4ed04b7c8metadatadisabled©show_logsîskip_as_script§cell_id$06e194ec-42dc-46aa-8935-86c4ed04b7c8code_foldedäcodeimg("accurate.jpg")$8f51a619-d0ad-42d2-bac9-acb9eab7cccbmetadatadisabled©show_logsîskip_as_script§cell_id$8f51a619-d0ad-42d2-bac9-acb9eab7cccbcode_foldedäcode-frametitle("Variational AutoEncoders (VAEs)")$d821d85c-74ae-44d0-909b-02d3099d8201metadatadisabled©show_logsîskip_as_script§cell_id$d821d85c-74ae-44d0-909b-02d3099d8201code_foldedäcodemd"""
``\mathcal{L}(x)`` is a **lower bound** to ``\log(f_X(x))`` as the Kullback-Leibler divergence ``D_\text{KL}((Y|X = x) \parallel (Z | X = x))`` is always nonnegative.
In the context of VAEs, ``Z`` represents the actual latent variable and ``Y`` represents the estimated random variables, so ``D_\text{KL}((Y|X = x) \parallel (Z | X = x))`` is a measure of the error made by our estimator.
"""$f50e757b-2f0d-458c-a577-3de499d9ba14metadatadisabled©show_logsîskip_as_script§cell_id$f50e757b-2f0d-458c-a577-3de499d9ba14code_foldedäcodemd"""
## Variational Auto-Encoder
We add artificial noise to improve learning:
```math
\mathbb{E}[\|X - D(E_\mu(X) + \mathcal{E} \odot E_\sigma(X))\|_2^2]
```
An insightful way to model this is as follows.
* Assume our dataset come from a distribution ``X`` that is obtained from a latent space ``Z`` using a decoder ``X = D(Z)``.
* If we knew the latent variable ``z``, we would search for a decoder ``D`` that maximizes ``\mathbb{E}[\log(f_{X|Z}(x|z))]``.
* Since we do not know ``z``, we make an estimate of ``y`` using an encoder ``Y = E(X)``.
* The validity of these encoder and decoder models can now be measured with a Maximum Likelihood Estimator that searches for the encoder and decoder models that maximize ``\mathbb{E}[\log(f_X(X))]``.
"""$c1fe952e-c074-49c1-91a4-0de76d235f25metadatadisabled©show_logsîskip_as_script§cell_id$c1fe952e-c074-49c1-91a4-0de76d235f25code_folded¤code(refs(args...) = bibrefs(biblio, args...)$530110cc-1dc9-427e-950b-64c0d04fc0dametadatadisabled©show_logsîskip_as_script§cell_id$530110cc-1dc9-427e-950b-64c0d04fc0dacode_foldedäcode&frametitle("Classifier-Free Guidance")$cddf4399-c659-4bbd-bae1-6d88f37a9b93metadatadisabled©show_logsîskip_as_script§cell_id$cddf4399-c659-4bbd-bae1-6d88f37a9b93code_foldedäcodeframetitle("Optical illusions")$2a0f227f-28b4-419e-9f82-f8bd13fca8c0metadatadisabled©show_logsîskip_as_script§cell_id$2a0f227f-28b4-419e-9f82-f8bd13fca8c0code_foldedäcodesection("Utils")$b94dd60a-2116-43ce-a5c3-5f1ff24479c0metadatadisabled©show_logsîskip_as_script§cell_id$b94dd60a-2116-43ce-a5c3-5f1ff24479c0code_foldedäcodeimg("stable_diffusion")$e33b16ec-4526-4dbe-affa-3e68aa508ab8metadatadisabled©show_logsîskip_as_script§cell_id$e33b16ec-4526-4dbe-affa-3e68aa508ab8code_foldedäcode8md"""
If ``\sigma`` is too large then ``\epsilon_\theta(y) \approx -\sigma \nabla_y \log f_{X+\sigma\mathcal{E}}(y)`` everywhere but the distribution ``Y \sim X + \sigma \mathcal{E}`` is too noisy, less specific to ``X`` (small signal to noise ratio). [Image source](https://yang-song.net/blog/2021/score/).
"""$2a728c9a-4268-40ff-905d-60b3b4c5b99bmetadatadisabled©show_logsîskip_as_script§cell_id$2a728c9a-4268-40ff-905d-60b3b4c5b99bcode_foldedäcode7refs(["ho2020Denoising", "permenter2024Interpretinga"])$de5ddee9-7a4c-4ae7-b6e2-ccdf1a406919metadatadisabled©show_logsîskip_as_script§cell_id$de5ddee9-7a4c-4ae7-b6e2-ccdf1a406919code_foldedäcodemd"""
Find encoder ``E`` and decoder ``D`` that minimize the loss:
```math
\mathbb{E}[\|X - D(E(X))\|_2^2]
```
The *code* (aka *latent variable*) ``z = E(x)`` typically has a smaller size compared to ``x`` to force the model to only keep essential features.
"""$df9e106b-16c3-4605-b5f6-0a4e82dd4b3bmetadatadisabled©show_logsîskip_as_script§cell_id$df9e106b-16c3-4605-b5f6-0a4e82dd4b3bcode_foldedäcodeHAlign(md"""
From Tweedie's formula, ``\epsilon`` is estimated to be
``-\sigma \nabla_y \log f_Y(y)``.
**Sampler** *Langevin dynamics*:
```math
\begin{multline}
y_{k+1} = y_k + \delta_k \nabla_y \log f_Y(y_k) + \sqrt{2\delta_k} w_k\\
\text{where } w_k \sim \mathcal{N}(0, 1)
\end{multline}
```
[Image source](https://yang-song.net/blog/2021/score/).
""",
img("langevin.gif", :width => 200),
)$e728cc9a-c17f-4143-9e0e-12c03a965becmetadatadisabled©show_logsîskip_as_script§cell_id$e728cc9a-c17f-4143-9e0e-12c03a965beccode_foldedäcode!refs("rombach2022HighResolution")$3fde5f41-32a9-4585-9b6a-131173947346metadatadisabled©show_logsîskip_as_script§cell_id$3fde5f41-32a9-4585-9b6a-131173947346code_folded¤codeZusing PlutoUI, DataFrames, PrettyTables, LinearAlgebra, Luxor, LaTeXStrings, MathTeXEngine$4b272be9-6881-41e8-9168-3fe471527aa8metadatadisabled©show_logsîskip_as_script§cell_id$4b272be9-6881-41e8-9168-3fe471527aa8code_folded¤code(import DocumenterCitations, CSV, Logging$44bb88d2-2fc7-4901-9fc3-890e54754932metadatadisabled©show_logsîskip_as_script§cell_id$44bb88d2-2fc7-4901-9fc3-890e54754932code_foldedäcodeframetitle("Tweedie's formula")$fbb03a1a-99ef-4d98-8ba4-10d3cc026abcmetadatadisabled©show_logsîskip_as_script§cell_id$fbb03a1a-99ef-4d98-8ba4-10d3cc026abccode_foldedäcode"refs("permenter2024Interpretinga")$784608dc-cf8f-483b-964f-b66575dc430emetadatadisabled©show_logsîskip_as_script§cell_id$784608dc-cf8f-483b-964f-b66575dc430ecode_foldedäcodeframetitle("Sampling")$1a1335db-6bc5-4a0e-a138-3513f7a2140dmetadatadisabled©show_logsîskip_as_script§cell_id$1a1335db-6bc5-4a0e-a138-3513f7a2140dcode_foldedäcode2md"""
```math
\begin{align}
\text{Auto-Encoder} & & D(E(X))\\
\text{Variational Auto-Encoder} & & D(E(X) + \mathcal{E})\\
\text{Denoising Auto-Encoder} & & D(E(X + \sigma \mathcal{E}))\\
\end{align}
```
The goal of the diffusion model is, given a noisy image ``Y = X + \sigma \mathcal{E}`` and ``\sigma``, to find the noise ``\mathcal{E}``.
The denoising Auto-Encoder instead attempts to find the original image ``X``.
At the limit ``\sigma \to 0``, the denoising Auto-Encoder ``D(E(X + \sigma \mathcal{E}))`` is a classical Auto-Encoder, hence the name.
To train the denoising Auto-Encoder, we can use the Evidence Lower-Bound with ``Y = X + \sigma \mathcal{E}`` and ``Z`` such that ``X = D(Z)`` $(cite("ho2020Denoising")):
```math
-\log(f_X(x)) \le \mathbb{E}[-\log(f_{X|Z}(x|Y))] + D((Y|X = x) \parallel Z)
```
"""$78dd39e3-d549-4012-92d1-3a1c8033ac29metadatadisabled©show_logsîskip_as_script§cell_id$78dd39e3-d549-4012-92d1-3a1c8033ac29code_foldedäcodeHAlign(
md"""
As ``\epsilon_\theta(x_t, \sigma_t)`` is more accurate than
``\epsilon_\theta(x_{t+1}, \sigma_{t+1})``,
$(cite("permenter2024Interpretinga", "Section 5"))
suggests accelerating the convergence by correcting part of
the previous step. That is, we take:
```math
\bar{\epsilon}_t = \textcolor{purple}{\gamma} \epsilon_\theta(x_t, \sigma_t) \textcolor{purple}{+ (1 - \gamma) \epsilon_\theta(x_{t+1}, \sigma_{t+1})}
```
with ``\gamma > 1``.
Image is $(cite("permenter2024Interpretinga", "Figure 3")).
""",
img("ge_step")
)$228d70f8-dd36-48a6-9c95-b2744c5b20b6metadatadisabled©show_logsîskip_as_script§cell_id$228d70f8-dd36-48a6-9c95-b2744c5b20b6code_foldedäcode3md"""
This can be approximated using Monte-Carlo given ``L`` samples ``\epsilon_1, \ldots \epsilon_L`` from the distribution ``\mathcal{N}(0, I)`` as
```math
\mathbb{E}[\log(f_{X|Z}(x|Y))] \approx \frac{1}{L} \sum_{i=1}^L \log(f_{X|Z}(x|E_\mu(x) + \epsilon_i \odot E_\sigma(x))).
```
In the simpler case where ``D_\sigma(z) = \mathbf{1}``, we recognize the classical L2 norm:
```math
\begin{align}
\mathbb{E}[\log(f_{X|Z}(x|Y))]
& \approx -\frac{\log(2\pi)}{2}+\frac{1}{L}\sum_{i=1}^L\|x - D_\mu(E_\mu(x) + \epsilon_i \odot E_\sigma(x))\|_2^2.
\end{align}
```
"""$76eaa13b-e76a-4e31-b702-78fd9694a059metadatadisabled©show_logsîskip_as_script§cell_id$76eaa13b-e76a-4e31-b702-78fd9694a059code_foldedäcode"frametitle("Monte-Carlo sampling")$ca8b4f75-f8c7-4384-80ae-fbbdba85bd46metadatadisabled©show_logsîskip_as_script§cell_id$ca8b4f75-f8c7-4384-80ae-fbbdba85bd46code_foldedäcode$frametitle("Denoising Auto-Encoder")$7d05bc54-a693-11ef-2d83-a5caeb91676emetadatadisabled©show_logsîskip_as_script§cell_id$7d05bc54-a693-11ef-2d83-a5caeb91676ecode_foldedäcodesection("Diffusion Models")$cd8567a7-f4fe-4592-ae4c-facdc35db563metadatadisabled©show_logsîskip_as_script§cell_id$cd8567a7-f4fe-4592-ae4c-facdc35db563code_folded¤codebiblio = load_biblio!()$97d6e0da-0b53-4467-95f3-a524575daa49metadatadisabled©show_logsîskip_as_script§cell_id$97d6e0da-0b53-4467-95f3-a524575daa49code_foldedäcodeimg("guidance")$c64fc228-d1e3-423f-8fb3-b8e7dff9d097metadatadisabled©show_logsîskip_as_script§cell_id$c64fc228-d1e3-423f-8fb3-b8e7dff9d097code_foldedäcoderefs("kingma2013AutoEncoding")$09829600-3488-492a-a507-53a4bbcb1cfcmetadatadisabled©show_logsîskip_as_script§cell_id$09829600-3488-492a-a507-53a4bbcb1cfccode_foldedäcode0refs(["burgert2023Diffusion", "geng2024Visual"])$76643525-2649-48b9-9d09-6d7dcef6e355metadatadisabled©show_logsîskip_as_script§cell_id$76643525-2649-48b9-9d09-6d7dcef6e355code_foldedäcode2md"""
Consider two deterministic functions ``E_\mu, E_\sigma : \mathbb{R}^n \to \mathbb{R}^r`` and ``D_\mu, D_\sigma : \mathbb{R}^r \to \mathbb{R}^n``.
Suppose ``X = D_\mu(Z) + \mathcal{E}_1 \odot D_\sigma(Z)`` and ``Y = E_\mu(X) + \mathcal{E}_2 \odot E_\sigma(X)`` with ``\mathcal{E}_1, \mathcal{E}_2 \sim \mathcal{N}(0, I)``.
We have (see $(cite("kingma2013AutoEncoding", "Appendix B")) for a proof):
```math
2D_\text{KL}((Y|X = x) \parallel Z) = \|E_\mu(X)\|_2^2 + \|E_\sigma(X)\|_2^2 - r - \sum_{i=1}^r \log((E_\sigma(X))_i^2)
```
For the second part of the ELBO, we have
```math
\begin{align}
& \mathbb{E}[\log(f_{X|Z}(x|Y))]\\
& = \mathbb{E}[\log(f_{X|Z}(x|E_\mu(x) + \mathcal{E}_2 \odot E_\sigma(x)))]\\
& = \mathbb{E}[\log(f_{\mathcal{E}_1}(\text{Diag}(D_\sigma(E_\mu(x) + \mathcal{E}_2 \odot E_\sigma(x)))^{-1} (x - D_\mu(E_\mu(x) + \mathcal{E}_2 \odot E_\sigma(x)))))]\\
& = -\frac{\log(2\pi)}{2}+\mathbb{E}[\|\text{Diag}(D_\sigma(E_\mu(x) + \mathcal{E}_2 \odot E_\sigma(x)))^{-1} (x - D_\mu(E_\mu(x) + \mathcal{E}_2 \odot E_\sigma(x)))\|_2^2].
\end{align}
```
"""$9cd90023-cf5d-4291-ae69-bc902c5c41bemetadatadisabled©show_logsîskip_as_script§cell_id$9cd90023-cf5d-4291-ae69-bc902c5c41becode_foldedäcode$HAlign(
md"""
It was shown in $(cite("burgert2023Diffusion")) how to train a diffusion model to generate illusions.
Suprisingly, $(cite("geng2024Visual")) showed that you don't need a specialized model and you can use a pre-trained diffusion model.
Given ``N`` transformations ``v_1, \ldots, v_N`` $(cite("geng2024Visual", "Equation (2)")):
```math
\bar{\epsilon}_t = \textcolor{purple}{\frac{1}{N} \sum_{i=1}^N v_i^{-1}(}\epsilon_\theta(\textcolor{purple}{v_i(}x_t\textcolor{purple}{)}, \sigma_t, \tau)\textcolor{purple}{)}
```
""",
img("flips")
)$0708fa36-bf99-44e3-a69e-991ad521d34bmetadatadisabled©show_logsîskip_as_script§cell_id$0708fa36-bf99-44e3-a69e-991ad521d34bcode_foldedäcoderefs("kingma2013AutoEncoding")$ca68ba1d-5f02-4f80-8cd1-d0c297f19f58metadatadisabled©show_logsîskip_as_script§cell_id$ca68ba1d-5f02-4f80-8cd1-d0c297f19f58code_foldedäcode#frametitle("Conditioned diffusion")$d3792580-6a04-4934-ad5a-cd054a5b4421metadatadisabled©show_logsîskip_as_script§cell_id$d3792580-6a04-4934-ad5a-cd054a5b4421code_foldedäcodeframetitle("Auto-Encoder")$8e39960f-ab0a-4103-bd6b-00e779333b01metadatadisabled©show_logsîskip_as_script§cell_id$8e39960f-ab0a-4103-bd6b-00e779333b01code_foldedäcodeqa(md"Proof",
md"""
> Reminder: ``D_\text{KL}(A \parallel B) = \sum_{a \in \text{Dom}(A)} f_A(a) \log \frac{f_A(a)}{f_B(a)} = \mathbb{E}[\log f_A(A) - \log f_B(A)]``
```math
\begin{align}
D_\text{KL}((Y|X = x) \parallel (Z | X = x))
& =
\mathbb{E}[\log(f_{Y|X}(Y|x)) - \log(f_{Z|X}(Y|x)) | X = x]\\
& =
\mathbb{E}[\log(f_{Y|X}(Y|x)) - \log(f_{Z,X}(Y,x)) | X = x]\\
& \qquad + \log(f_{X}(x))\\
& =
\mathbb{E}[\log(f_{Y|X}(Y|x)) - \log(f_{X|Z}(x|Y)) - \log(f_{Z}(Y)) | X = x]\\
& \qquad + \log(f_{X}(x))\\
& =
D_\text{KL}((Y|X = x) \parallel Z) - \mathbb{E}[\log(f_{X|Z}(x|Y)) | X = x]\\
& \qquad + \log(f_{X}(x))
\end{align}
```
""",
)$87f3dafc-6452-4a3f-b635-9b7497e1481ametadatadisabled©show_logsîskip_as_script§cell_id$87f3dafc-6452-4a3f-b635-9b7497e1481acode_foldedäcodeimg("DDIM")last_hot_reload_time cell_dependencies @$0e01c57f-65b1-480c-a258-9a76513ac8d6precedence_heuristic cell_id$0e01c57f-65b1-480c-a258-9a76513ac8d6downstream_cells_mapupstream_cells_map@md_strgetindexcite$d077a471-6b43-41df-8218-e3b4a7e38550$6fd4e448-992f-4976-82e9-6313ba038d26precedence_heuristic cell_id$6fd4e448-992f-4976-82e9-6313ba038d26downstream_cells_mapupstream_cells_map@md_strgetindexcite$d077a471-6b43-41df-8218-e3b4a7e38550$a9e46466-cdbf-4539-ae36-a3113634b353precedence_heuristic cell_id$a9e46466-cdbf-4539-ae36-a3113634b353downstream_cells_mapupstream_cells_mapframetitle$3cdcd14a-9102-4ed4-b186-629ce183b73cprecedence_heuristic cell_id$3cdcd14a-9102-4ed4-b186-629ce183b73cdownstream_cells_mapupstream_cells_mapimg$abc07fcf-35f5-43eb-90aa-23ec0b581e56precedence_heuristic 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Waiting for notebook process to start... Done. Starting precompilation...PlutoUI
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Waiting for notebook process to start... Done. Starting precompilation...LinearAlgebra
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Waiting for notebook process to start... Done. Starting precompilation...Luxor
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Waiting for notebook process to start... Done. Starting precompilation...DataFrames
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Waiting for notebook process to start... Done. Starting precompilation...waiting_for_permission·restart_recommended_msgrestart_required_msginstalled_versionsLoggingstdlibMathTeXEngine0.6.9PlutoUI0.7.83PrettyTables3.4.8LaTeXStrings1.4.1!__internal_julia_manifest_version1.13.3LinearAlgebrastdlibDocumenterCitations1.5.0__internal_julia_version1.13.3CSV1.0.0DataFrames1.8.2Luxor4.5.0install_time_ns>~instantiatedípluto_versionv1.1.0in_temp_dir