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paper/paper.md

@@ -114,19 +114,17 @@ where the score function $\nabla_{\boldsymbol{x}}\log p_t(\boldsymbol{x})$ is su
 
 The score-based diffusion model for the data is fit by optimising the parameters of the network $\theta$ via stochastic gradient descent of the score-matching loss  
 
-$$
-    % \mathbb{E}_{t\sim\mathcal{U}(0, T)}\mathbb{E}_{\boldsymbol{x}\sim p(\boldsymbol{x})}[\lambda(t)||\nabla_{\boldsymbol{x}}\log p_t(\boldsymbol{x}) - \boldsymbol{s}_{\theta}(\boldsymbol{x},t)||_2^2]
-
+<!-- $$
     % \mathcal{L}(\theta) = \mathbb{E}_{t\sim\mathcal{U}(0, T)}\mathbb{E}_{\boldsymbol{x}\sim p(\boldsymbol{x})}\mathbb{E}_{\boldsymbol{x}(t)\sim p(\boldsymbol{x}(t)|\boldsymbol{x})}[\lambda(t)||\nabla_{\boldsymbol{x}}\log p_t(\boldsymbol{x}(t)|\boldsymbol{x}(0)) - \boldsymbol{s}_{\theta}(\boldsymbol{x}(t),t)||_2^2]
     x
-$$
+$$ -->
 
 where $\lambda(t)$ is an arbitrary scalar weighting function, chosen to preferentially weight certain times - usually near $t=0$ where the data has only a small amount of noise added. Here, $p_t(\boldsymbol{x}(t)|\boldsymbol{x}(0))$ is the transition kernel for Gaussian diffusion paths. This is defined depending on the form of the SDE [@sde] and for the common variance-preserving (VP) SDE the kernel is written as 
 
-$$
+<!-- $$
     % p(\boldsymbol{x}(t)|\boldsymbol{x}(0)) = \mathcal{G}[\boldsymbol{x}(t)|\mu_t \cdot \boldsymbol{x}(0), \sigma^2_t \cdot \mathbb{I}]
     x
-$$
+$$ -->
 
 where $\mathcal{G}[\cdot]$ is a Gaussian distribution, $\mu_t=\exp(-\int_0^t\text{d}s \; \beta(s))$ and $\sigma^2_t = 1 - \mu_t$. $\beta(t)$ is typically chosen to be a simple linear function of $t$.