typos

lecture12.md

@@ -511,7 +511,7 @@ class: middle
 
 The training objective for $s\_\theta(\mathbf{x}\_t, t)$ is then a weighted sum of Fisher divergences for all noise levels $t$, 
 $$\sum\_{t=1}^T \lambda(t) \mathbb{E}\_{p\_{t}(\mathbf{x}\_t)} \left[ || \nabla\_{\mathbf{x}\_t} \log p\_{t}(\mathbf{x}\_t) - s\_\theta(\mathbf{x}\_t, t) ||\_2^2 \right]$$
-where $\lambda(t)$ is a weighting function that increases with $t$ to give more importance to the noisier samples.
+where $\lambda(t)$ is a weighting function.
 
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@@ -530,12 +530,12 @@ class: middle
 ## Interpretation 3: Denoising score matching
 
 A third interpretation of VDMs can be obtained by reparameterizing $\mathbf{x}\_0$ using Tweedie's formula, as
-$$\mathbf{x}\_0 = \frac{\mathbf{x}\_t + (1-\bar{\alpha}\_t) \nabla\_{\mathbf{x}\_t} \log q(\mathbf{x}\_t | \mathbf{x}\_0) }{\sqrt{\bar{\alpha}\_t}},$$
+$$\mathbf{x}\_0 = \frac{\mathbf{x}\_t + (1-\bar{\alpha}\_t) \nabla\_{\mathbf{x}\_t} \log q(\mathbf{x}\_t) }{\sqrt{\bar{\alpha}\_t}},$$
 which we can plug into the the mean of the tractable posterior to obtain
 $$\begin{aligned}
 \mu\_q(\mathbf{x}\_t, \mathbf{x}\_0, t) &= \frac{\sqrt{\alpha\_t}(1-\bar{\alpha}\_{t-1})}{1-\bar{\alpha}\_t}\mathbf{x}\_t + \frac{\sqrt{\bar{\alpha}\_{t-1}}(1-\alpha\_t)}{1-\bar{\alpha}\_t}\mathbf{x}\_0 \\\\
 &= ... \\\\
-&= \frac{1}{\sqrt{\alpha}\_t} \mathbf{x}\_t + \frac{1-\alpha\_t}{\sqrt{\alpha\_t}} \nabla\_{\mathbf{x}\_t} \log q(\mathbf{x}\_t | \mathbf{x}\_0).
+&= \frac{1}{\sqrt{\alpha}\_t} \mathbf{x}\_t + \frac{1-\alpha\_t}{\sqrt{\alpha\_t}} \nabla\_{\mathbf{x}\_t} \log q(\mathbf{x}\_t).
 \end{aligned}$$
 
 ???
@@ -552,17 +552,19 @@ $$\mu\_\theta(\mathbf{x}\_t, t) = \frac{1}{\sqrt{\alpha}\_t} \mathbf{x}\_t + \fr
 Under this parameterization, the minimization of the expected KL divergence $L\_{t-1}$ can be rewritten as
 $$\begin{aligned}
 &\arg \min\_\theta \mathbb{E}\_{q(\mathbf{x}\_t | \mathbf{x}\_0)}\text{KL}(q(\mathbf{x}\_{t-1}|\mathbf{x}\_t, \mathbf{x}\_0) || p\_\theta(\mathbf{x}\_{t-1} | \mathbf{x}\_t) )\\\\
-=&\arg \min\_\theta \mathbb{E}\_{q(\mathbf{x}\_t | \mathbf{x}\_0)} \frac{1}{2\sigma^2\_t} \frac{(1-\alpha\_t)^2}{\alpha\_t} || s\_\theta(\mathbf{x}\_t, t) - \nabla\_{\mathbf{x}\_t}  \log q(\mathbf{x}\_t | \mathbf{x}\_0) ||_2^2
+=&\arg \min\_\theta \mathbb{E}\_{q(\mathbf{x}\_t | \mathbf{x}\_0)} \frac{1}{2\sigma^2\_t} \frac{(1-\alpha\_t)^2}{\alpha\_t} || s\_\theta(\mathbf{x}\_t, t) - \nabla\_{\mathbf{x}\_t}  \log q(\mathbf{x}\_t) ||_2^2
 \end{aligned}$$
 
-.success[Optimizing a score-based model amounts to learning a neural network that predicts the score $\nabla\_{\mathbf{x}\_t} \log q(\mathbf{x}\_t | \mathbf{x}\_0)$  of the tractable posterior.]
+.success[Optimizing a score-based model amounts to learning a neural network that predicts the score $\nabla\_{\mathbf{x}\_t} \log q(\mathbf{x}\_t)$.]
 
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 class: middle
 
-Since $s\_\theta(\mathbf{x}\_t, t)$ is learned in expectation over the data distribution $q(\mathbf{x}\_0)$, the score network will eventually approximate the score of the marginal distribution $q(\mathbf{x}\_t$), for each noise level $t$, that is
-$$s\_\theta(\mathbf{x}\_t, t) \approx \nabla\_{\mathbf{x}\_t} \log q(\mathbf{x}\_t).$$
+Unfortunately, $\nabla\_{\mathbf{x}\_t} \log q(\mathbf{x}\_t)$ is not tractable in general. 
+However, since $s\_\theta(\mathbf{x}\_t, t)$ is learned in expectation over the data distribution $q(\mathbf{x}\_0)$, minimizing instead
+$$\mathbb{E}\_{q(\mathbf{x}\_0)} \mathbb{E}\_{q(\mathbf{x}\_t | \mathbf{x}\_0)} \frac{1}{2\sigma^2\_t} \frac{(1-\alpha\_t)^2}{\alpha\_t} || s\_\theta(\mathbf{x}\_t, t) - \nabla\_{\mathbf{x}\_t}  \log q(\mathbf{x}\_t | \mathbf{x}\_0) ||\_2^2$$
+ensures that $s\_\theta(\mathbf{x}\_t, t) \approx \nabla\_{\mathbf{x}\_t} \log q(\mathbf{x}\_t)$.
 
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