Diffusion prior and measurement scores

Let $p_{\sigma }$, $Q_{\sigma }$, $m_{\sigma }$, and $r_{\sigma }(y\mid x)$ be formed under the hypotheses of the conditional-score and Tweedie identities for Gaussian smoothing. The diffusion prior score is $s_{\sigma }(x)={\mathrm{\nabla }}_x\log p_{\sigma }(x)$, the exact score of the smoothed prior density. The measurement score is $g_{\sigma }(x,y)={\mathrm{\nabla }}_x\log r_{\sigma }(y\mid x)$, where $r_{\sigma }(y\mid x)=\int {\mathrm{\ell }}_y(u)Q_{\sigma }(x,du)$ is the likelihood density averaged over the conditional law of the clean state. For general inverse problems, evaluating this clean-state likelihood integral is generally computationally intractable. The conditional score is $c_{\sigma }(x,y)={\mathrm{\nabla }}_x\log p_{\sigma }(x\mid y)$; it equals $s_{\sigma }(x)+g_{\sigma }(x,y)$, and the clean-state conditional mean is $m_{\sigma }(x)=x+{\sigma }^2{s}_{\sigma }(x)$. A learned field ${\widehat{s}}_{\theta }$ is an estimate of $s_{\sigma }$ and is exact only when it agrees with $s_{\sigma }$ almost everywhere under $p_{\sigma }(x)dx$. Replacing $r_{\sigma }(y\mid x)$ by the likelihood evaluated at $m_{\sigma }(x)$ is a plug-in likelihood approximation, not part of the exact measurement-score definition.

Conditional-score and Tweedie identities for Gaussian smoothing

Daras et al., §§2.2 and 2.4.3–2.4.4

Song et al., §5

Chung et al., §3.1