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and the term inside becomes a least squares regression, so if the network actually reaches the global minimum of loss, then we have <math>\epsilon_\theta(x_t, t) = \frac{x_t -\sqrt{\bar\alpha_t} E_q[x_0|x_t]}{\sigma_t} = -\sigma_t\nabla_{x_t}\ln q(x_t)</math>
Thus, a score-based network predicts noise, and can be used for denoising
Conversely, the continuous limit <math>x_{t-1} = x_{t-dt}, \beta_t = \beta(t) dt, z_t\sqrt{dt} = dW_t</math> of the backward equation
<math display="block">x_{t-1} = \frac{x_t}{\sqrt{\alpha_t}}- \frac{ \beta_t}{\sigma_{t}\sqrt{\alpha_t }} \epsilon_\theta(x_t, t) + \sqrt{\beta_t} z_t; \quad z_t \sim \mathcal{N}(0, I)</math>
gives us precisely the same equation as score-based diffusion:
<math display="block">x_{t-dt} = x_t(1+\beta(t)dt / 2) + \beta(t) \nabla_{x_t}\ln q(x_t) dt + \sqrt{\beta(t)}dW_t</math>Thus,
== Main variants ==
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