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<h1>Denoising Diffusion Probabilistic Models (DDPM)</h1>
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<p>This is a <a href="https://pytorch.org">PyTorch</a> implementation/tutorial of the paper
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<a href="https://papers.labml.ai/paper/2006.11239">Denoising Diffusion Probabilistic Models</a>.</p>
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<p>In simple terms, we get an image from data and add noise step by step.
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Then We train a model to predict that noise at each step and use the model to
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generate images.</p>
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<p>The following definitions and derivations show how this works.
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For details please refer to <a href="https://papers.labml.ai/paper/2006.11239">the paper</a>.</p>
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<h2>Forward Process</h2>
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<p>The forward process adds noise to the data $x_0 \sim q(x_0)$, for $T$ timesteps.</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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q(x_t | x_{t-1}) = \mathcal{N}\big(x_t; \sqrt{1- \beta_t} x_{t-1}, \beta_t \mathbf{I}\big) \\
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q(x_{1:T} | x_0) = \prod_{t = 1}^{T} q(x_t | x_{t-1})
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\end{align}</script>
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</p>
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<p>where $\beta_1, \dots, \beta_T$ is the variance schedule.</p>
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<p>We can sample $x_t$ at any timestep $t$ with,</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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q(x_t|x_0) &= \mathcal{N} \Big(x_t; \sqrt{\bar\alpha_t} x_0, (1-\bar\alpha_t) \mathbf{I} \Big)
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\end{align}</script>
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</p>
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<p>where $\alpha_t = 1 - \beta_t$ and $\bar\alpha_t = \prod_{s=1}^t \alpha_s$</p>
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<h2>Reverse Process</h2>
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<p>The reverse process removes noise starting at $p(x_T) = \mathcal{N}(x_T; \mathbf{0}, \mathbf{I})$
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for $T$ time steps.</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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\color{cyan}{p_\theta}(x_{t-1} | x_t) &= \mathcal{N}\big(x_{t-1};
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\color{cyan}{\mu_\theta}x_t, t), \color{cyan}{\Sigma_\theta}(x_t, t)\big) \\
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\color{cyan}{p_\theta}(x_{0:T}) &= \color{cyan}{p_\theta}(x_T) \prod_{t = 1}^{T} \color{cyan}{p_\theta}(x_{t-1} | x_t) \\
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\color{cyan}{p_\theta}(x_0) &= \int \color{cyan}{p_\theta}(x_{0:T}) dx_{1:T}
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\end{align}</script>
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</p>
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<p>$\color{cyan}\theta$ are the parameters we train.</p>
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<h2>Loss</h2>
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<p>We optimize the ELBO (from Jenson’s inequality) on the negative log likelihood.</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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\mathbb{E}[-\log \color{cyan}{p_\theta}(x_0)]
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&\le \mathbb{E}_q [ -\log \frac{\color{cyan}{p_\theta}(x_{0:T})}{q(x_{1:T}|x_0)} ] \\
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&=L
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\end{align}</script>
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</p>
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<p>The loss can be rewritten as follows.</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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L
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&= \mathbb{E}_q [ -\log \frac{\color{cyan}{p_\theta}(x_{0:T})}{q(x_{1:T}|x_0)} ] \\
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&= \mathbb{E}_q [ -\log p(x_T) - \sum_{t=1}^T \log \frac{\color{cyan}{p_\theta}(x_{t-1}|x_t)}{q(x_t|x_{t-1})} ] \\
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&= \mathbb{E}_q [
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-\log \frac{p(x_T)}{q(x_T|x_0)}
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-\sum_{t=2}^T \log \frac{\color{cyan}{p_\theta}(x_{t-1}|x_t)}{q(x_{t-1}|x_t,x_0)}
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-\log \color{cyan}{p_\theta}(x_0|x_1)] \\
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&= \mathbb{E}_q [
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D_{KL}(q(x_T|x_0) \Vert p(x_T))
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+\sum_{t=2}^T D_{KL}(q(x_{t-1}|x_t,x_0) \Vert \color{cyan}{p_\theta}(x_{t-1}|x_t))
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-\log \color{cyan}{p_\theta}(x_0|x_1)]
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\end{align}</script>
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</p>
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<p>$D_{KL}(q(x_T|x_0) \Vert p(x_T))$ is constant since we keep $\beta_1, \dots, \beta_T$ constant.</p>
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<h3>Computing $L_{t-1} = D_{KL}(q(x_{t-1}|x_t,x_0) \Vert \color{cyan}{p_\theta}(x_{t-1}|x_t))$</h3>
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<p>The forward process posterior conditioned by $x_0$ is,</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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q(x_{t-1}|x_t, x_0) &= \mathcal{N} \Big(x_{t-1}; \tilde\mu_t(x_t, x_0), \tilde\beta_t \mathbf{I} \Big) \\
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\tilde\mu_t(x_t, x_0) &= \frac{\sqrt{\bar\alpha_{t-1}}\beta_t}{1 - \bar\alpha_t}x_0
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+ \frac{\sqrt{\alpha_t}(1 - \bar\alpha_{t-1})}{1-\bar\alpha_t}x_t \\
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\tilde\beta_t &= \frac{1 - \bar\alpha_{t-1}}{a}
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\end{align}</script>
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</p>
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<p>The paper sets $\color{cyan}{\Sigma_\theta}(x_t, t) = \sigma_t^2 \mathbf{I}$ where $\sigma_t^2$ is set to constants
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$\beta_t$ or $\tilde\beta_t$.</p>
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<p>Then,
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<script type="math/tex; mode=display">\color{cyan}{p_\theta}(x_{t-1} | x_t) = \mathcal{N}\big(x_{t-1}; \color{cyan}{\mu_\theta}(x_t, t), \sigma_t^2 \mathbf{I} \big)</script>
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</p>
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<p>For given noise $\epsilon \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ using $q(x_t|x_0)$</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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x_t(x_0, \epsilon) &= \sqrt{\bar\alpha_t} x_0 + \sqrt{1-\bar\alpha_t}\epsilon \\
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x_0 &= \frac{1}{\sqrt{\bar\alpha_t}} \Big(x_t(x_0, \epsilon) - \sqrt{1-\bar\alpha_t}\epsilon\Big)
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\end{align}</script>
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</p>
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<p>This gives,</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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L_{t-1}
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&= D_{KL}(q(x_{t-1}|x_t,x_0) \Vert \color{cyan}{p_\theta}(x_{t-1}|x_t)) \\
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&= \mathbb{E}_q \Bigg[ \frac{1}{2\sigma_t^2}
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\Big \Vert \tilde\mu(x_t, x_0) - \color{cyan}{\mu_\theta}(x_t, t) \Big \Vert^2 \Bigg] \\
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&= \mathbb{E}_{x_0, \epsilon} \Bigg[ \frac{1}{2\sigma_t^2}
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\bigg\Vert \frac{1}{\sqrt{\alpha_t}} \Big(
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x_t(x_0, \epsilon) - \frac{\beta_t}{\sqrt{1 - \bar\alpha_t}} \epsilon
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\Big) - \color{cyan}{\mu_\theta}(x_t(x_0, \epsilon), t) \bigg\Vert^2 \Bigg] \\
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\end{align}</script>
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</p>
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<p>Re-parameterizing with a model to predict noise</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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\color{cyan}{\mu_\theta}(x_t, t) &= \tilde\mu \bigg(x_t,
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\frac{1}{\sqrt{\bar\alpha_t}} \Big(x_t -
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\sqrt{1-\bar\alpha_t}\color{cyan}{\epsilon_\theta}(x_t, t) \Big) \bigg) \\
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&= \frac{1}{\sqrt{\alpha_t}} \Big(x_t -
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\frac{\beta_t}{\sqrt{1-\bar\alpha_t}}\color{cyan}{\epsilon_\theta}(x_t, t) \Big)
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\end{align}</script>
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</p>
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<p>where $\epsilon_theta$ is a learned function that predicts $\epsilon$ given $(x_t, t)$.</p>
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<p>This gives,</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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L_{t-1}
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&= \mathbb{E}_{x_0, \epsilon} \Bigg[ \frac{\beta_t^2}{2\sigma_t^2 \alpha_t (1 - \bar\alpha_t)}
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\Big\Vert
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\epsilon - \color{cyan}{\epsilon_\theta}(\sqrt{\bar\alpha_t} x_0 + \sqrt{1-\bar\alpha_t}\epsilon, t)
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\Big\Vert^2 \Bigg]
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\end{align}</script>
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</p>
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<p>That is, we are training to predict the noise.</p>
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<h3>Simplified loss</h3>
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<p>
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<script type="math/tex; mode=display">L_simple(\theta) = \mathbb{E}_{t,x_0, \epsilon} \Bigg[ \bigg\Vert
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\epsilon - \color{cyan}{\epsilon_\theta}(\sqrt{\bar\alpha_t} x_0 + \sqrt{1-\bar\alpha_t}\epsilon, t)
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\bigg\Vert^2 \Bigg]</script>
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</p>
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<p>This minimizes $-\log \color{cyan}{p_\theta}(x_0|x_1)$ when $t=1$ and $L_{t-1}$ for $t\gt1$ discarding the
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weighting in $L_{t-1}$. Discarding the weights $\frac{\beta_t^2}{2\sigma_t^2 \alpha_t (1 - \bar\alpha_t)}$
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increase the weight given to higher $t$ (which have higher noise levels), therefore increasing the sample quality.</p>
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<p>This file implements the loss calculation and a basic sampling method that we use to generate images during
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training.</p>
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<p>Here is the <a href="unet.html">UNet model</a> that gives $\color{cyan}{\epsilon_\theta}(x_t, t)$ and
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<a href="experiment.html">training code</a>.
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<a href="evaluate.html">This file</a> can generate samples and interpolations from a trained model.</p>
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<p><a href="https://app.labml.ai/run/a44333ea251411ec8007d1a1762ed686"><img alt="View Run" src="https://img.shields.io/badge/labml-experiment-brightgreen" /></a></p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">162</span><span></span><span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Optional</span>
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<span class="lineno">163</span>
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<span class="lineno">164</span><span class="kn">import</span> <span class="nn">torch</span>
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<span class="lineno">165</span><span class="kn">import</span> <span class="nn">torch.nn.functional</span> <span class="k">as</span> <span class="nn">F</span>
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<span class="lineno">166</span><span class="kn">import</span> <span class="nn">torch.utils.data</span>
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<span class="lineno">167</span><span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
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<span class="lineno">168</span>
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<span class="lineno">169</span><span class="kn">from</span> <span class="nn">labml_nn.diffusion.ddpm.utils</span> <span class="kn">import</span> <span class="n">gather</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-1'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-1'>#</a>
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</div>
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<h2>Denoise Diffusion</h2>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">172</span><span class="k">class</span> <span class="nc">DenoiseDiffusion</span><span class="p">:</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-2'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-2'>#</a>
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</div>
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<ul>
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<li><code>eps_model</code> is $\color{cyan}{\epsilon_\theta}(x_t, t)$ model</li>
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<li><code>n_steps</code> is $t$</li>
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<li><code>device</code> is the device to place constants on</li>
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</ul>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">177</span> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eps_model</span><span class="p">:</span> <span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">device</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">device</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-3'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-3'>#</a>
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</div>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">183</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
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<span class="lineno">184</span> <span class="bp">self</span><span class="o">.</span><span class="n">eps_model</span> <span class="o">=</span> <span class="n">eps_model</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-4'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-4'>#</a>
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</div>
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<p>Create $\beta_1, \dots, \beta_T$ linearly increasing variance schedule</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">187</span> <span class="bp">self</span><span class="o">.</span><span class="n">beta</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.0001</span><span class="p">,</span> <span class="mf">0.02</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-5'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-5'>#</a>
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</div>
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<p>$\alpha_t = 1 - \beta_t$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">190</span> <span class="bp">self</span><span class="o">.</span><span class="n">alpha</span> <span class="o">=</span> <span class="mf">1.</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">beta</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-6'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-6'>#</a>
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</div>
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<p>$\bar\alpha_t = \prod_{s=1}^t \alpha_s$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">192</span> <span class="bp">self</span><span class="o">.</span><span class="n">alpha_bar</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cumprod</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">alpha</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-7'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-7'>#</a>
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</div>
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<p>$T$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">194</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_steps</span> <span class="o">=</span> <span class="n">n_steps</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-8'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-8'>#</a>
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|
</div>
|
|
<p>$\sigma^2 = \beta$</p>
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|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">196</span> <span class="bp">self</span><span class="o">.</span><span class="n">sigma2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">beta</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-9'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-9'>#</a>
|
|
</div>
|
|
<h4>Get $q(x_t|x_0)$ distribution</h4>
|
|
<p>
|
|
<script type="math/tex; mode=display">\begin{align}
|
|
q(x_t|x_0) &= \mathcal{N} \Big(x_t; \sqrt{\bar\alpha_t} x_0, (1-\bar\alpha_t) \mathbf{I} \Big)
|
|
\end{align}</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">198</span> <span class="k">def</span> <span class="nf">q_xt_x0</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x0</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">t</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">)</span> <span class="o">-></span> <span class="n">Tuple</span><span class="p">[</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">]:</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-10'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-10'>#</a>
|
|
</div>
|
|
<p><a href="utils.html">gather</a> $\alpha_t$ and compute $\sqrt{\bar\alpha_t} x_0$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">208</span> <span class="n">mean</span> <span class="o">=</span> <span class="n">gather</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">alpha_bar</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span> <span class="o">**</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">x0</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-11'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-11'>#</a>
|
|
</div>
|
|
<p>$(1-\bar\alpha_t) \mathbf{I}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">210</span> <span class="n">var</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">gather</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">alpha_bar</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-12'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-12'>#</a>
|
|
</div>
|
|
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">212</span> <span class="k">return</span> <span class="n">mean</span><span class="p">,</span> <span class="n">var</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-13'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-13'>#</a>
|
|
</div>
|
|
<h4>Sample from $q(x_t|x_0)$</h4>
|
|
<p>
|
|
<script type="math/tex; mode=display">\begin{align}
|
|
q(x_t|x_0) &= \mathcal{N} \Big(x_t; \sqrt{\bar\alpha_t} x_0, (1-\bar\alpha_t) \mathbf{I} \Big)
|
|
\end{align}</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">214</span> <span class="k">def</span> <span class="nf">q_sample</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x0</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">t</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">eps</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-14'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-14'>#</a>
|
|
</div>
|
|
<p>$\epsilon \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">224</span> <span class="k">if</span> <span class="n">eps</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
|
|
<span class="lineno">225</span> <span class="n">eps</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn_like</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-15'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-15'>#</a>
|
|
</div>
|
|
<p>get $q(x_t|x_0)$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">228</span> <span class="n">mean</span><span class="p">,</span> <span class="n">var</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">q_xt_x0</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-16'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-16'>#</a>
|
|
</div>
|
|
<p>Sample from $q(x_t|x_0)$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">230</span> <span class="k">return</span> <span class="n">mean</span> <span class="o">+</span> <span class="p">(</span><span class="n">var</span> <span class="o">**</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">*</span> <span class="n">eps</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-17'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-17'>#</a>
|
|
</div>
|
|
<h4>Sample from $\color{cyan}{p_\theta}(x_{t-1}|x_t)$</h4>
|
|
<p>
|
|
<script type="math/tex; mode=display">\begin{align}
|
|
\color{cyan}{p_\theta}(x_{t-1} | x_t) &= \mathcal{N}\big(x_{t-1};
|
|
\color{cyan}{\mu_\theta}(x_t, t), \sigma_t^2 \mathbf{I} \big) \\
|
|
\color{cyan}{\mu_\theta}(x_t, t)
|
|
&= \frac{1}{\sqrt{\alpha_t}} \Big(x_t -
|
|
\frac{\beta_t}{\sqrt{1-\bar\alpha_t}}\color{cyan}{\epsilon_\theta}(x_t, t) \Big)
|
|
\end{align}</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">232</span> <span class="k">def</span> <span class="nf">p_sample</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">xt</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">t</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-18'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-18'>#</a>
|
|
</div>
|
|
<p>$\color{cyan}{\epsilon_\theta}(x_t, t)$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">246</span> <span class="n">eps_theta</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eps_model</span><span class="p">(</span><span class="n">xt</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-19'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-19'>#</a>
|
|
</div>
|
|
<p><a href="utils.html">gather</a> $\bar\alpha_t$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">248</span> <span class="n">alpha_bar</span> <span class="o">=</span> <span class="n">gather</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">alpha_bar</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-20'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-20'>#</a>
|
|
</div>
|
|
<p>$\alpha_t$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">250</span> <span class="n">alpha</span> <span class="o">=</span> <span class="n">gather</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">alpha</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-21'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-21'>#</a>
|
|
</div>
|
|
<p>$\frac{\beta}{\sqrt{1-\bar\alpha_t}}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">252</span> <span class="n">eps_coef</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">alpha</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">alpha_bar</span><span class="p">)</span> <span class="o">**</span> <span class="o">.</span><span class="mi">5</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-22'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-22'>#</a>
|
|
</div>
|
|
<p>
|
|
<script type="math/tex; mode=display">\frac{1}{\sqrt{\alpha_t}} \Big(x_t -
|
|
\frac{\beta_t}{\sqrt{1-\bar\alpha_t}}\color{cyan}{\epsilon_\theta}(x_t, t) \Big)</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">255</span> <span class="n">mean</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="n">alpha</span> <span class="o">**</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">xt</span> <span class="o">-</span> <span class="n">eps_coef</span> <span class="o">*</span> <span class="n">eps_theta</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-23'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-23'>#</a>
|
|
</div>
|
|
<p>$\sigma^2$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">257</span> <span class="n">var</span> <span class="o">=</span> <span class="n">gather</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">sigma2</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-24'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-24'>#</a>
|
|
</div>
|
|
<p>$\epsilon \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">260</span> <span class="n">eps</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">xt</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">xt</span><span class="o">.</span><span class="n">device</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-25'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-25'>#</a>
|
|
</div>
|
|
<p>Sample</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">262</span> <span class="k">return</span> <span class="n">mean</span> <span class="o">+</span> <span class="p">(</span><span class="n">var</span> <span class="o">**</span> <span class="o">.</span><span class="mi">5</span><span class="p">)</span> <span class="o">*</span> <span class="n">eps</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-26'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-26'>#</a>
|
|
</div>
|
|
<h4>Simplified Loss</h4>
|
|
<p>
|
|
<script type="math/tex; mode=display">L_simple(\theta) = \mathbb{E}_{t,x_0, \epsilon} \Bigg[ \bigg\Vert
|
|
\epsilon - \color{cyan}{\epsilon_\theta}(\sqrt{\bar\alpha_t} x_0 + \sqrt{1-\bar\alpha_t}\epsilon, t)
|
|
\bigg\Vert^2 \Bigg]</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">264</span> <span class="k">def</span> <span class="nf">loss</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x0</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">noise</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-27'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-27'>#</a>
|
|
</div>
|
|
<p>Get batch size</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">273</span> <span class="n">batch_size</span> <span class="o">=</span> <span class="n">x0</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-28'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-28'>#</a>
|
|
</div>
|
|
<p>Get random $t$ for each sample in the batch</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">275</span> <span class="n">t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_steps</span><span class="p">,</span> <span class="p">(</span><span class="n">batch_size</span><span class="p">,),</span> <span class="n">device</span><span class="o">=</span><span class="n">x0</span><span class="o">.</span><span class="n">device</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">long</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-29'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-29'>#</a>
|
|
</div>
|
|
<p>$\epsilon \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">278</span> <span class="k">if</span> <span class="n">noise</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
|
|
<span class="lineno">279</span> <span class="n">noise</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn_like</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-30'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-30'>#</a>
|
|
</div>
|
|
<p>Sample $x_t$ for $q(x_t|x_0)$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">282</span> <span class="n">xt</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">q_sample</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">noise</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-31'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-31'>#</a>
|
|
</div>
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<p>Get $\color{cyan}{\epsilon_\theta}(\sqrt{\bar\alpha_t} x_0 + \sqrt{1-\bar\alpha_t}\epsilon, t)$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">284</span> <span class="n">eps_theta</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eps_model</span><span class="p">(</span><span class="n">xt</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-32'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-32'>#</a>
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</div>
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<p>MSE loss</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">287</span> <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">mse_loss</span><span class="p">(</span><span class="n">noise</span><span class="p">,</span> <span class="n">eps_theta</span><span class="p">)</span></pre></div>
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