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correct typo diffusion
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mlelarge committed Sep 15, 2023
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2 changes: 1 addition & 1 deletion modules/18a-diffusion.md
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Expand Up @@ -72,7 +72,7 @@ q(x_{t-1}|x_t,x_0) = \mathcal{N}(x_{t-1};\mu(x_t,x_0), \gamma_t I),
\end{align*}
with
\begin{align*}
\mu(x_t,x_0) &= \frac{\sqrt{\alpha_t}(1-\overline{\alpha}_{t-1})}{1-\overline{\alpha}_{t}}x_t + \frac{\sqrt{\overline{\alpha}_{t-1}\beta_t}}{1-\overline{\alpha}_{t}}x_0\\
\mu(x_t,x_0) &= \frac{\sqrt{\alpha_t}(1-\overline{\alpha}_{t-1})}{1-\overline{\alpha}_{t}}x_t + \frac{\beta_t\sqrt{\overline{\alpha}_{t-1}}}{1-\overline{\alpha}_{t}}x_0\\
\gamma_t &= \frac{1-\overline{\alpha}_{t-1}}{1-\overline{\alpha}_{t}}\beta_t
\end{align*}
but we know that $x_0 = 1/\sqrt{\overline{\alpha}_t}\left( x_t-\sqrt{1-\overline{\alpha}_t}\epsilon\right)$, hence we have
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