Learned inverse estimators and reconstruction targets

Let $(X,d_X)$ and $(Y,d_Y)$ be metric parameter and data spaces, let $U\subseteq X$ be a nonempty admissible truth class, let $F:U\to Y$ be a forward map, and let $(Q,d_Q)$ be a metric target space. All metric spaces in this definition carry their Borel sigma-algebras. Give $U$ the subspace Borel sigma-algebra $\mathcal{B}(U):=\{U\cap A:A\in \mathcal{B}(X)\}$, and assume that $q:U\to Q$ is measurable from $(U,\mathcal{B}(U))$ to $(Q,\mathcal{B}(Q))$. The target $q(x)$ is the recovery object assigned to truth $x$; $Q$ may be the parameter space itself or a quotient solution-class space. A point reconstruction rule is a Borel map $g:Y\to Q$, and its output at data $y$ is $g(y)$. A learned point estimator is such a map ${\widehat{q}}_{\varphi }$ with a fitted parameter $\varphi$ in a specified parameter set. A complete training description names the probability law $P_{\mathrm{f}\mathrm{i}\mathrm{t}}$ of all variables used to fit $\varphi$; those variables may be paired or unpaired. A complete test description names a probability law $P_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}$ for evaluation pairs $(Y,q(X))$. No equality between fitting and test laws is assumed.

A finite unrolled learned update consists of an integer $K\ge 1$, a Borel initialization $u_0:Y\to Q$, and maps $T_{\varphi ,k}:Q\times Y\to Q$ measurable from $(Q\times Y,\mathcal{B}(Q)\otimes \mathcal{B}(Y))$ to $(Q,\mathcal{B}(Q))$ for $0\le k<K$, with $u_{k+1}(y)=T_{\varphi ,k}(u_k(y),y)$. Here the layer domain uses the product sigma-algebra, not an assumed identification with the Borel sigma-algebra of the product topology. Its output $u_K:Y\to Q$ is a point reconstruction rule: if $u_k$ is Borel, the preimage of a measurable rectangle $A\times C$ under $y\mapsto (u_k(y),y)$ is $u_k^{-1}(A)\cap C$, so this pairing is product-measurable; composing it with $T_{\varphi ,k}$ and applying finite induction proves that every $u_k$ is Borel. An amortized inverse map is a learned point estimator applied to multiple data inputs after a single fit; this term adds a reuse convention, not an accuracy guarantee.

For a real Hilbert signal space $H$, a denoiser is a Borel map $D_{\varphi }:H\to H$ trained under a specified signal-corruption law and used to transform a corrupted signal into a signal estimate or an update. A denoiser is the exact minimum mean-square error map only when it equals the conditional mean for the specified signal and corruption laws. A learned prior is a supplied Borel probability measure ${\rho }_{\varphi }$ on $H$; one construction is the pushforward law ${\rho }_{\varphi }(A)=\nu (G_{\varphi }^{-1}(A))$ for $A\in \mathcal{B}(H)$, where $\nu$ is a specified probability measure on a measurable source space $(Z,\mathcal{Z})$ and $G_{\varphi }:(Z,\mathcal{Z})\to (H,\mathcal{B}(H))$ is measurable. The law or generator alone is not a data-to-target estimator.

For a finite horizon $T>0$, a specified parameter set $\mathrm{\Phi }$, and $\varphi \in \mathrm{\Phi }$, a score field model in a finite-dimensional signal space ${\mathbb{R}}^d$ is a Borel map $s_{\varphi }:[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ intended to approximate the target field $s^{\ast }(t,x)={\mathrm{\nabla }}_x\log p_t(x)$ for a specified family $(p_t{)}_{t\in [0,T]}$ of positive continuously differentiable probability densities for which $s^{\ast }$ is jointly Borel. Its complete description names a probability law $P_{\mathrm{f}\mathrm{i}\mathrm{t}}$ for all variables used to fit $\varphi$ and a probability law $P_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}$ on $[0,T]\times {\mathbb{R}}^d$ for evaluation inputs $(t,x)$; the target on those inputs is $s^{\ast }(t,x)$. These laws need not agree, and no approximation guarantee under either law is implied. A score field is not itself a posterior law, posterior sample, or point reconstruction rule. A posterior sampler is a probability kernel $S:Y\to Q$; relative to a supplied posterior target kernel ${\mathrm{\Pi }}_y$, it is exact on a data domain $D$ when $S_y={\mathrm{\Pi }}_y$ as probability measures for every $y\in D$. A draw from $S_y$ is random; obtaining a point estimate from it requires an additional summary rule. The name “learned” by itself asserts no exact recovery, stability, convergence, calibration, or train-to-test guarantee.

Metric parameter/data model, admissible truths, forward map, and target-class map

Deterministic data-only estimator interface and its exactness convention

Evidence-normalized inverse posterior and data-indexed kernel convention

Real Hilbert signal spaces are complete real inner-product spaces

Measurable spaces and their sigma-algebras

Probability measures used for fitting and signal laws

Borel measurability of reconstruction maps

Probability kernels for posterior samplers

Product-measurable layer inputs for finite unrolling

Ongie et al., §§3.1 and 4.1–4.2.1, full paper

Daras et al., §§1.2 and 2.4.2–2.4.3, full survey

Dashti and Stuart, §3.2, full lecture notes