Open frontiers in calibrated and robust learned inversion

Fix integers $d,n\ge 1$, parameters $p,s,a,R,\sigma >0$, a credibility error $\gamma \in (0,1)$, and a continuous schedule $\beta :[0,T]\to (0,\mathrm{\infty })$ with $T>0$. Let ${\mathrm{\Theta }}_s(R)=\{\theta \in {\mathbb{R}}^d:{\sum }_{j=1}^d{j}^{2s}{\theta }_j^2\le R^2\},A_p=\mathrm{diag}(1^{-p},2^{-p},\dots ,d^{-p})\mathrm{.}$ Observe $Y_i=A_p\theta +{\epsilon }_i$, $1\le i\le n$, with independent ${\epsilon }_i\sim N(0,{\sigma }^2{I}_d)$. Use the proper Gaussian prior ${\mathrm{\Pi }}_0=N(0,\mathrm{diag}(j^{-1-2a}{)}_{j=1}^d)$, and let ${\mathrm{\Pi }}_n(\cdot \mid Y)$ be its exact posterior under the stated forward operator and noise law. Let $m_n(Y)$ be the posterior mean and let $r_{n,\gamma }(Y)$ be the unique radius for which the Euclidean ball $C_{n,\gamma }(Y)=\{u\in {\mathbb{R}}^d:\mathrm{\parallel }u-m_n(Y){\mathrm{\parallel }}_2\le r_{n,\gamma }(Y)\}$ has posterior probability $1-\gamma$. Define the exact frequentist coverage $c_{n,\gamma }(\theta )=P_{n,\theta }\{\theta \in C_{n,\gamma }(Y)\}$ and its baseline error ${\delta }_{n,\gamma }^0={\sup }_{\theta \in {\mathrm{\Theta }}_s(R)}\mathrm{\mid }{c}_{n,\gamma }(\theta )-(1-\gamma )\mathrm{\mid }$.

For a deterministic numerical matrix ${\widehat{A}}_n\in {\mathbb{R}}^{d\times d}$ with ${\delta }_A\ge 0$ and $\mathrm{\parallel }{\widehat{A}}_n-A_p{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {\delta }_A$, let ${\widehat{\mathrm{\Pi }}}_n(\cdot \mid Y)$ be the Gaussian posterior formed with ${\widehat{A}}_n$ and the same prior and noise variance. Write ${\widehat{p}}_{n,t}^Y$ for the density obtained by applying the forward variance-preserving diffusion $d{X}_t=-\frac{1}{2}\beta (t)X_{t}dt+\sqrt{\beta (t)}d{W}_t$ to $X_0\sim {\widehat{\mathrm{\Pi }}}_n(\cdot \mid Y)$, and set ${\widehat{s}}_{n,t}^Y=\mathrm{\nabla }\log {\widehat{p}}_{n,t}^Y$. A jointly Borel learned score field $s_{n,t}^Y$ is required to satisfy ${\sup }_{\theta \in {\mathrm{\Theta }}_s(R)}{E}_{Y\sim P_{n,\theta }}[\frac{1}{T}{\int }_0^T{\int }_{{\mathbb{R}}^d}\mathrm{\parallel }{s}_{n,t}^Y(x)-{\widehat{s}}_{n,t}^Y(x){\mathrm{\parallel }}_2^2{\widehat{p}}_{n,t}^Y(x)dxdt]\le {\epsilon }_s^2\mathrm{.}$ For $h=T\mathrm{/}K$, define the reverse-time Euler chain by $Z_0\sim {\widehat{p}}_{n,T}^Y$ and, for $k=0,\dots ,K-1$, $Z_{k+1}=Z_k+h[\frac{1}{2}\beta (T-kh)Z_k+\beta (T-kh)s_{n,T-kh}^Y(Z_k)]+\sqrt{h\beta (T-kh)}{\zeta }_k,$ where the ${\zeta }_k$ are independent $N(0,I_d)$ vectors independent of $Z_0$ and the observations. Draw $M$ independent copies of this chain conditionally on $Y$ and denote their terminal values by $X^{(1)},\dots ,X^{(M)}$. Let $\bar{X}=M^{-1}{\sum }_i{X}^{(i)}$, let $\widehat{r}$ be the empirical $1-\gamma$ quantile of $\{\mathrm{\parallel }{X}^{(i)}-\bar{X}{\mathrm{\parallel }}_2{\}}_{i=1}^M$ (the smallest order statistic whose empirical cumulative mass is at least $1-\gamma$), and put ${\widehat{C}}_{n,K,M}(Y)=\{u:\mathrm{\parallel }u-\bar{X}{\mathrm{\parallel }}_2\le \widehat{r}\}$. Suppose the exact credible-ball boundary satisfies the uniform margin condition ${\sup }_{\theta \in {\mathrm{\Theta }}_s(R)}{P}_{n,\theta }(\mid \mathrm{\parallel }\theta -m_n(Y){\mathrm{\parallel }}_2-r_{n,\gamma }(Y)\mid \le u)\le C_{\mathrm{a}\mathrm{c}}{u}^{\kappa }(0<u\le u_0)$ for fixed $C_{\mathrm{a}\mathrm{c}},u_0>0$ and $\kappa >0$. Determine whether a minimax-sharp nonasymptotic function $\mathrm{\Phi }$ exists uniformly over all such score fields and, if it does, find it as a function of $d,n,p,s,a,R,\sigma ,T,\gamma ,C_{\mathrm{a}\mathrm{c}},u_0,\kappa ,{\delta }_A,{\epsilon }_s,K^{-1}$, and $M^{-1\mathrm{/}2}$ such that ${\sup }_{\theta \in {\mathrm{\Theta }}_s(R)}\mid P_{n,\theta ,\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{r}}\{\theta \in {\widehat{C}}_{n,K,M}(Y)\}-(1-\gamma )\mid \le {\delta }_{n,\gamma }^0+\mathrm{\Phi },$ with matching lower bounds that identify when $\mathrm{\Phi }\to 0$. The probability includes observation noise and sampler randomness. The target concerns the additional coverage error introduced by the approximate forward operator, learned score, finite reverse discretization, and finite posterior sample; it does not assume that the exact Bayesian credible ball is already calibrated on every truth class.

Knapik, van der Vaart, and van Zanten, full paper

Szabó, van der Vaart, and van Zanten, full paper

Giné and Nickl, full textbook

Narnhofer et al., full paper

As of 2026-10-19, the full original minimax question is not resolved by this review. A specific obstruction is now proved: with $d=n=R=\sigma =T=1$, $\beta =1$, $\gamma =1\mathrm{/}4$, and zero numerical-operator and integrated score errors, changing the score only at the deterministic Euler grid times can make empirical-ball coverage at truth zero tend to zero along $M=K\to \mathrm{\infty }$. The required uniform boundary-margin condition still holds, and the exact credible-ball baseline error is strictly smaller than $3\mathrm{/}4$. Every valid bound therefore has a positive limiting additional-error term along this admissible scalar path; zero integrated score error and a refined solver grid do not by themselves imply vanishing additional coverage error.

This is a partial answer, not a solution of the full parameter-dependent minimax problem. The original question asks for a sharp function and the regimes in which it vanishes; it does not require it to vanish along every path on which the listed approximation parameters tend to zero. Constant upper bounds exist because every coverage probability lies in $[0,1]$. The proved scalar obstruction does not determine the optimal bound across all original dimensions, sample sizes, credibility levels, margins and error parameters, or characterize every possible vanishing regime. Those parts of the unchanged original formulation remain unanswered here.

The cited Gaussian inverse-sequence papers address exact credible-set coverage and prior/truth dependence. Narnhofer et al.'s conformal error-quantile coverage concerns a different set and does not supply the requested Bayesian credible-ball perturbation bound. No global novelty claim or formulation-specific solution is inferred from the absence of that joint result in these sources.

A scalar zero-integrated-error obstruction to uniform decay along a refined Euler grid. ↗

Let $\mathbb{T}=\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ carry normalized Lebesgue measure and fix the real orthonormal Fourier basis ${\varphi }_1(x)=1,{\varphi }_{2k}(x)=\sqrt{2}\cos (kx),{\varphi }_{2k+1}(x)=\sqrt{2}\sin (kx)(k\ge 1)\mathrm{.}$ Use this basis to identify the real Hilbert space ${\mathrm{\ell }}^2={\mathrm{\ell }}^2({\mathbb{N}}_{>0};\mathbb{R})$, where ${\mathbb{N}}_{>0}=\{1,2,\dots \}$, with $L^2(\mathbb{T};\mathbb{R})$. Fix integers $N,m\ge 1$ and parameters $s,p,R,\sigma >0$, and define ${\mathrm{\Theta }}_s(R)=\{\theta \in {\mathrm{\ell }}^2:{\sum }_{j\ge 1}{j}^{2s}{\theta }_j^2\le R^2\},(A_p\theta {)}_j=j^{-p}{\theta }_j,{\mathcal{X}}_p(R)=A_p{\mathrm{\Theta }}_s(R)\mathrm{.}$ The inverse map $G^{†}:{\mathcal{X}}_p(R)\to {\mathrm{\ell }}^2$ is $G^{†}(A_p\theta )=\theta$. For $m\ge 1$, let $P_m$ retain the first $m$ Fourier coefficients and let $I_m:{\mathbb{R}}^m\to {\mathrm{\ell }}^2$ append zero coordinates.

Fix a globally Lipschitz nonpolynomial function ${\sigma }_0:\mathbb{R}\to \mathbb{R}$, called the activation. For $x\in {\mathbb{R}}^m$, form the scalar input field ${\bar{u}}_0(x;z)={\sum }_{j=1}^m{x}_j{\varphi }_j(z)$. A pointwise affine lift $u_0(x;z)=V_{\mathrm{i}\mathrm{n}}{\bar{u}}_0(x;z)+b_{\mathrm{i}\mathrm{n}}$, with $V_{\mathrm{i}\mathrm{n}}\in {\mathbb{R}}^{c_0}$ and $b_{\mathrm{i}\mathrm{n}}\in {\mathbb{R}}^{c_0}$, creates $c_0$ channels. For fields $u_{\mathrm{\ell }}(z)\in {\mathbb{R}}^{c_{\mathrm{\ell }}}$ with positive integer channel counts $c_{\mathrm{\ell }}$, apply an integer number $D\ge 1$ of layers indexed by $\mathrm{\ell }=0,\dots ,D-1$: $u_{\mathrm{\ell }+1}(z)={\sigma }_0\text{\hspace{0.17em}⁣}(W_{\mathrm{\ell }}{u}_{\mathrm{\ell }}(z)+b_{\mathrm{\ell }}+({\mathcal{K}}_{\mathrm{\ell }}{u}_{\mathrm{\ell }})(z))\mathrm{.}$ Here ${\sigma }_0$ acts coordinatewise, $W_{\mathrm{\ell }}\in {\mathbb{R}}^{c_{\mathrm{\ell }+1}\times c_{\mathrm{\ell }}}$ and $b_{\mathrm{\ell }}\in {\mathbb{R}}^{c_{\mathrm{\ell }+1}}$ are pointwise channel weights, and ${\mathcal{K}}_{\mathrm{\ell }}$ is a Fourier multiplier meaning $\widehat{({\mathcal{K}}_{\mathrm{\ell }}u)}(k)=R_{\mathrm{\ell },k}\widehat{u}(k)$, where $\widehat{u}(k)=(2\pi {)}^{-1}{\int }_0^{2\pi }u(z)e^{-ikz}dz$ is computed componentwise, with matrices $R_{\mathrm{\ell },k}\in {\mathbb{C}}^{c_{\mathrm{\ell }+1}\times c_{\mathrm{\ell }}}$ satisfying $R_{\mathrm{\ell },-k}=\overline{R_{\mathrm{\ell },k}}$ and $R_{\mathrm{\ell },k}=0$ for $\mathrm{\mid }k\mathrm{\mid }>q$ for some integer cutoff $q\ge 0$. A pointwise affine output map $v(x;z)=a_{\mathrm{o}\mathrm{u}\mathrm{t}}^{\mathrm{\top }}{u}_D(z)+b_{\mathrm{o}\mathrm{u}\mathrm{t}}$ gives a real scalar field, and the network output is ${\mathrm{\Psi }}_m(x)=(⟨v(x;\cdot ),{\varphi }_j{⟩}_{L^2}{)}_{j=1}^m$. Let ${\mathcal{F}}_{S,B}^{(m)}$ be the union of these maps over architectures whose depth, channel counts, Fourier cutoff, and number of real scalar parameters are each at most the integer $S\ge 1$, with every trainable parameter bounded in absolute value by $B>0$. A parameter array used at mesh size $m$ is the same array at every other mesh size. For fixed $S$, there are finitely many allowed architecture indices; equip their disjoint union of bounded real parameter boxes with its Borel structure.

Write $\mathcal{P}(K)$ for the Borel probability laws on a Borel set $K$. For a Borel probability law $\mu$ on ${\mathrm{\Theta }}_s(R)$, draw independent training truths ${\theta }_i\sim \mu$ and observations $X_{i,m}=P_m{A}_p{\theta }_i+{\eta }_{i,m},Y_{i,m}=P_m{\theta }_i,{\eta }_{i,m}\sim N(0,{\sigma }^2{m}^{-1}{I}_m),i=1,\dots ,N,$ with all noises independent, and write $D_{N,m}=((X_{i,m},Y_{i,m}){)}_{i=1}^N$. A selector is Borel into the finite disjoint union of architecture indices and their bounded parameter boxes, and its realized map minimizes the empirical loss. For each $N,m,S,B$ with ${\mathcal{F}}_{S,B}^{(m)}\mathrm{\ne }\mathrm{\varnothing }$, assume there is at least one such selector $\tau$ that, from $D_{N,m}$, returns a minimizer of $\frac{1}{N}{\sum }_{i=1}^N\mathrm{\parallel }\mathrm{\Psi }(X_{i,m})-Y_{i,m}{\mathrm{\parallel }}_2^2$ over $\mathrm{\Psi }\in {\mathcal{F}}_{S,B}^{(m)}$; let ${\mathcal{T}}_{N,m}(S,B)$ be the set of all such selectors, and set it to $\mathrm{\varnothing }$ when ${\mathcal{F}}_{S,B}^{(m)}=\mathrm{\varnothing }$. At test time, draw an independent $\theta \sim \mu$ and ${\eta }_m\sim N(0,{\sigma }^2{m}^{-1}{I}_m)$, and define ${\mathfrak{R}}_{N,m}^{\mathrm{E}\mathrm{R}\mathrm{M}}(S,B)={\inf }_{\tau \in {\mathcal{T}}_{N,m}(S,B)}{\sup }_{\mu \in \mathcal{P}({\mathrm{\Theta }}_s(R))}E{\parallel I_m\tau (D_{N,m})(P_m{A}_p\theta +{\eta }_m)-\theta \parallel }_{{\mathrm{\ell }}^2}^2\mathrm{.}$ Use the convention $\inf \mathrm{\varnothing }=+\mathrm{\infty }$, so an empty architecture class has infinite ERM risk. Also define ${\mathfrak{R}}_{N,m}^{\mathrm{a}\mathrm{l}\mathrm{l}}$ by the same infimum over all Borel predictors from $(D_{N,m},X_m)$ to ${\mathbb{R}}^m$, with $X_m=P_m{A}_p\theta +{\eta }_m$. Determine matching minimax upper and lower rates for ${\mathfrak{R}}_{N,m}^{\mathrm{E}\mathrm{R}\mathrm{M}}(S,B)$ jointly in $N,m,S,B,p,s,R,\sigma$. In particular, characterize the joint growth regimes of $m=m_N$ and $S=S_N$ that separate statistical estimation, neural-operator approximation, and continuum truncation error, and determine when the bounded shared-parameter Fourier architecture attains the unrestricted rate ${\mathfrak{R}}_{N,m}^{\mathrm{a}\mathrm{l}\mathrm{l}}$.

Reinhardt, Wang, and Zech, full 2026 JMLR paper

de Hoop et al., full EIT inverse-map paper

Bal, full inverse-problems lecture notes

Kovachki et al., full 2023 JMLR paper

As of 2026-10-19, the original full joint $N,m,S,B$ risk-rate question is not resolved by this review. One obstruction is now explicit: the specified coordinate-free Fourier architecture is translation-equivariant, whereas $A_p^{-1}$ rescales the first cosine and sine differently. For every odd mesh $m\ge 3$, its worst-law risk has the positive floor $R^2(3^p-2^p{)}^2\mathrm{/}(43^{2(p+s)})$, independently of training size and parameter budget, while an unrestricted cutoff predictor has risk tending to zero. Thus unrestricted-rate attainment along odd meshes is ruled out for this original architecture. The corresponding all-parameter bounded-class rate, including even meshes, is not supplied by that argument.

Reinhardt, Wang, and Zech give function-space regression/ERM bounds under their stated compactness, entropy and noise hypotheses, including a holomorphic FrameNet application; these are not the original noisy-input, joint-mesh inverse-risk rate. Kovachki et al. describe Fourier layers, translation symmetries and continuum approximation with their stated architecture conventions. The EIT inverse-map question concerns a generalized architecture with coordinate/chart lifting and a different compact dataset, so this coordinate-free obstruction does not identify the two formulations. No global novelty claim or claim that every finite-sample neural inverse problem is unsolved is made.

Explicit odd-mesh obstruction and unrestricted comparison. ↗

The statement is a mesh-dependent extension of the existing draft question on finite-sample generalization of neural inverse maps. It does not use that open question as a premise or supplier.

Related draft question on finite-sample generalization without a mesh-rate claim. ‡