Measurement-guided reverse-kernel reconstruction

Let $N\ge 1$ be finite, let $0=t_0<t_1<\cdots <t_N$, let $d\ge 1$ and let the state space be ${\mathbb{R}}^d$, and let $(\mathsf{Y},\mathcal{Y})$ be a measurable observation space. Fix a measurable forward operator $\mathcal{A}:{\mathbb{R}}^d\to \mathsf{Y}$, an observation $y\in \mathsf{Y}$, and a nonnegative measurable likelihood function ${\mathrm{\ell }}_y:{\mathbb{R}}^d\to [0,\mathrm{\infty })$ obtained by fixing $y$ in a likelihood density relative to a fixed sigma-finite measure on $\mathsf{Y}$. A measurement-guided reverse-kernel reconstruction procedure takes a finite Borel score field ${\widehat{s}}_{\theta }:{\mathbb{R}}^d\times \{t_0,\dots ,t_N\}\to {\mathbb{R}}^d$; a terminal reference probability ${\nu }_N$ on the Borel sets of ${\mathbb{R}}^d$ and a sampler that returns an exact draw or failure within finite initialization budget $b_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}$; and, for each $k$, a measurable correction space $({\mathsf{C}}_k,{\mathcal{C}}_k)$, its failure-augmented measurable space ${\mathsf{C}}_k^{†}={\mathsf{C}}_k\bigsqcup \{\mathrm{\perp }\}$ with $\sigma$-algebra ${\mathcal{C}}_k^{†}=\{D,D\cup \{\mathrm{\perp }\}:D\in {\mathcal{C}}_k\}$, a measurable correction map $C_{k,{\widehat{s}}_{\theta },\mathcal{A},{\mathrm{\ell }}_y}:{\mathbb{R}}^d\times \mathsf{Y}\to {\mathsf{C}}_k^{†}$, a probability kernel $Q_{\theta ,k}(\cdot \mid x,y,a)$ from ${\mathbb{R}}^d\times \mathsf{Y}\times {\mathsf{C}}_k$ with its product sigma-algebra to ${\mathbb{R}}^d$ with its Borel sigma-algebra, and a sampler that returns an exact draw from that kernel or failure within finite per-step budget $b_k$. The correction map may use the supplied score, forward operator and likelihood and, when supplied for the exact-score option, the prior and scale map. It may also take, for each step, a declared Borel map ${\mathrm{\Pi }}_{k-1}:{\mathbb{R}}^d\to {\mathbb{R}}^d$ and a Borel admissible set $A_{k-1}\subseteq {\mathbb{R}}^d$. Each per-step budget includes correction evaluation, kernel sampling and any declared projection. First draw $x_N$ from ${\nu }_N$ within $b_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}$; if initialization fails, return the terminal-initialization failure status. For $k=N,N-1,\dots ,1$, compute $a_k=C_{k,{\widehat{s}}_{\theta },\mathcal{A},{\mathrm{\ell }}_y}(x_k,y)$; if $a_k=\mathrm{\perp }$, return the correction failure status at $k$; otherwise draw $x_{k-1}$ from $Q_{\theta ,k}(\cdot \mid x_k,y,a_k)$ within $b_k$; if drawing fails, return the sampling failure status at $k$; if ${\mathrm{\Pi }}_{k-1}$ is supplied, apply it to $x_{k-1}$ and return the admissibility failure status at $k-1$ if evaluation fails or its output is outside $A_{k-1}$. If $k>1$, continue at $k-1$ with the resulting state $x_{k-1}$; if $k=1$, return $x_0$ with completed status. A correction may use a declared surrogate without additional likelihood or noising assumptions. If it uses the exact measurement score at step $k$, the caller must also declare a Borel probability measure $\mu$ on ${\mathbb{R}}^d$ and a map $\sigma :\{t_1,\dots ,t_N\}\to (0,\mathrm{\infty })$ associating each reverse-schedule time with its Gaussian-noising scale. For every such step, the forward noising convention at $t_k$ must be $X_{\sigma (t_k)}=X_0+\sigma (t_k)Z$, where $X_0\sim \mu$ and $Z\sim N(0,I_d)$ is independent of $X_0$, and the observation variable $Y$ with conditional density ${\mathrm{\ell }}_y$ must be conditionally independent of $Z$ given $X_0$, and ${\mathrm{\ell }}_y$ must be bounded, strictly positive and Borel measurable with $0<Z_y=\int {\mathrm{\ell }}_y(u)\mu (du)<\mathrm{\infty }$. The exact correction then uses $g_{\sigma (t_k)}(x_k,y)={\mathrm{\nabla }}_x\log \int {\mathrm{\ell }}_y(u)Q_{\sigma (t_k)}(x_k,du)$, with $Q_{\sigma (t_k)}$ formed from this prior and Gaussian noising as in the conditional-score and Tweedie identities. The procedure returns an algorithmic sample; it makes no posterior-exactness claim unless the terminal law and all transition and projection conditions are separately certified.

diagram failed to render:
\begin{tikzpicture}[>=stealth]
  \node[draw,rounded corners,align=center] (init) at (0,0) {draw $x_N\sim\nu_N$ within $b_{\mathrm{init}}$};
  \node[draw,rounded corners,align=center] (corr) at (3.5,0) {$a_k=C_k(x_k,y)$; check $a_k\ne\bot$};
  \node[draw,rounded corners,align=center] (kernel) at (7.5,0) {draw $x_{k-1}\sim Q_{\theta,k}(\cdot\mid x_k,y,a_k)$};
  \node[draw,rounded corners,align=center] (proj) at (11.5,0) {optional $\Pi_{k-1}$; check $A_{k-1}$};
  \node[draw,diamond,aspect=2,align=center] (next) at (15.3,0) {$k=1?$};
  \node[draw,rounded corners,align=center] (out) at (15.3,-1.8) {return $x_0$};
  \node[draw,rounded corners,align=center] (stop) at (7.5,-2.2) {first failure status};
  \draw[->] (init)--(corr);
  \draw[->] (corr)--(kernel);
  \draw[->] (kernel)--(proj);
  \draw[->] (proj)--(next);
  \draw[->] (next.south)--node[right]{$k=1$}(out.north);
  \draw[->] (next.north) -- ++(0,1.5) -| node[pos=0.25,above,align=center]{$k>1$: continue at $k-1$\\with the just-produced state}(corr.north);
  \draw[->] (init.south)--(stop.north);
  \draw[->] (corr.south)--(stop.north);
  \draw[->] (kernel.south)--(stop.north);
  \draw[->] (proj.south)--(stop.north);
\end{tikzpicture}

The upper return arrow repeats the reverse step with the newly produced state and decremented index. The downward arrows record immediate stopping at a failed initialization, correction, kernel draw, or declared projection. The diagram is an operational flow description and asserts no distributional identity.

Diffusion prior and measurement scores

Measure kernel and probability kernel

Daras et al., §§2.4 and 3

Song et al., §5 and predictor-corrector algorithms

Chung et al., §3.1 and Algorithms 1–2

Prince, Chapter 18, “Diffusion models”