Coordinate-free Fourier inverse maps have an odd-mesh risk floor

Consider the complete original Fourier architecture and diagonal inverse question reproduced below. For every odd integer $m\ge 3$, every $N\ge 1$, and every admissible bounded parameter class with a measurable empirical minimizer, its risk satisfies ${\mathfrak{R}}_{N,m}^{\mathrm{E}\mathrm{R}\mathrm{M}}(S,B)\ge \frac{R^2(3^p-2^p{)}^2}{43^{2(p+s)}}>0.$ The same bound holds for every measurable learner whose realized predictor belongs to the stated coordinate-free Fourier class, whether or not it minimizes empirical risk. On the other hand, the unrestricted risk tends to zero as $m\to \mathrm{\infty }$ for fixed $p,s,R,\sigma >0$, independently of $N$. Thus no growth of $N,S,B$ can make this original architecture attain the vanishing unrestricted risk along an odd-mesh subsequence. This is a partial resolution: no complete joint bounded-class rate, or corresponding fixed positive even-mesh floor, is asserted. The original question is retained under Original question with explicit positive coordinate indices and empty-class conventions.

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

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 ${\mathrm{\ell }}^2({\mathbb{N}}_{>0})$ 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 ${\mathfrak{R}}_{N,m}^{\mathrm{a}\mathrm{l}\mathrm{l}}$.

The odd-mesh consistency subquestion is answered negatively, and the unrestricted vanishing comparison is proved. The full requested joint N,m,S,B minimax rate is not resolved here; in particular even meshes break the complete-pair symmetry used by the lower bound, and different activations/parameter budgets may have distinct approximation behavior. This is not the EIT question: the generalized EIT architecture has coordinate/chart lifting, so this coordinate-free translation obstruction is not transferred to it.