Constructive Universal Approximation by Infinite-Dimensional Neural Networks via Gaussian-Weighted Ridgelet Transform

Sho Sonoda; Luca Galimberti; Yuanchao Xu; Isao Ishikawa; Masahiro Ikeda

cs.LGOct 16, 2026

Constructive Universal Approximation by Infinite-Dimensional Neural Networks via Gaussian-Weighted Ridgelet Transform

Authors: Sho Sonoda, Luca Galimberti, Yuanchao Xu, Isao Ishikawa, Masahiro Ikeda

Organizations: RIKEN Center for Advanced Intelligence Project (AIP), Nihonbashi 1-chome Mitsui Building, 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan · AI tech studio, CyberAgent, Inc., Shibuya Scramble Square, 2-24-12 Shibuya, Shibuya-ku, Tokyo 150-6121, Japan · Department of Mathematics, King’s College London, Strand Building, London, WC2R 2LS, UK · Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan ·

Abstract

Neural networks whose inputs are functions, as in operator learning, are usually shown to be universal by existence arguments, either by reduction to finite-dimensional networks or by Stone--Weierstrass-type density; these prove that approximants exist but say little about how the parameters depend on the target. We take a constructive route. We extend the ridgelet transform, an integral transform related to the Fourier, Radon, and wavelet transforms, to continuous maps on a separable infinite-dimensional Hilbert space, replacing the Lebesgue measure that is unavailable in infinite dimensions by Gaussian weights with a homogeneous mixture on the direction space. The resulting transform has a Plancherel identity and reconstruction formulas incorporating the frame operator, the correction induced by the weights, and it assigns to each target an explicit coefficient measure, one for every admissible analysis filter. From these representations we obtain uniform approximation on compact sets for all continuous non-polynomial activations of polynomial growth, including ReLU, and, by Monte Carlo sampling of the coefficient, finite-width networks with a dimension-free $O(N^{-1/2})$ error in the width $N$ for globally Lipschitz activations and coefficients with finite variation and second moments. Explicit examples include Gaussian targets and neural-operator layers. A companion Lean~4 formalization provides machine-checked proofs, with its correspondence to the manuscript documented at https://shosonoda.github.io/lean-operator-ridgelet/ .

Figures & tables

Experiment	$d$	error at $N=4$	error at $N=4096$	slope
1: ReLU, Gaussian directions on $\ell^{2}$	$10$	$0.41$	$0.011$	$-0.53$
	$10^{2}$	$0.37$	$0.012$	$-0.50$
	$10^{3}$	$0.42$	$0.011$	$-0.51$
	$10^{4}$	$0.32$	$0.010$	$-0.50$
2: ridgelet coefficient of $f_{W}$ , target $6\,f_{g}$	$10$	$11.9$	$0.46$	$-0.47$
	$10^{2}$	$11.6$	$0.43$	$-0.48$

Table 1: Mean supremum error on

K

of the sampled networks at

N=4

and

N=4096

, and the fitted slope of the error against

N

in logarithmic coordinates, for each truncation dimension

d

Panel of Figure 1 — Figure 1: Mean supremum error on $K$ against the width $N$ for Experiments 1 (upper left), 2 (upper right), and 3(a) (bottom), for several truncation dimensions $d$ ; the dashed line is proportional to $N^{-1/2}$ .

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Reference	Target class	Approximant	Norm or topology
Chen and Chen [19]	continuous functionals on a compact $K\subset C[a,b]$ or $L^{p}[a,b]$	one hidden layer, Tauber–Wiener activation applied to finitely many point values or integrals of the input	uniform on $K$
Chen and Chen [20]	continuous operators $G\colon V\to C(K_{2})$ , $V\subset C(K_{1})$ compact	branch–trunk network (the DeepONet form), Tauber–Wiener activations	uniform on $V\times K_{2}$
Stinchcombe [91]	continuous functions on compactifications; functionals through the same structure	one hidden layer, activation characterized by the compactification	uniform on the compactification
Rossi and Conan-Guez [81]	continuous functionals on a compact $K\subset L^{2}(V)$	one hidden layer of functional neurons $\rho(\int wx+b)$ , regular activation	uniform on $K$
Benth et al. [9]	continuous $f\colon X\to\mathbb{R}$ , $X$ Fréchet	one hidden layer $\sum_{i}\langle\ell_{i},\sigma(A_{i}x+b_{i})\rangle$ , operator activation $\sigma\colon X\to X$ bounded and separating	cc
Galimberti [32] (Thm. 4.2)	continuous functions on quasi-Polish spaces	same architecture composed with an injection into $\ell^{2}$ ; finite-rank projections (Thm. 4.3)	cc

Appendix

Table 2: Universal approximation with infinite-dimensional inputs: functionals, Fréchet and topological inputs, weighted spaces, and signatures.

Reference	Target class	Approximant	Norm or topology
Kovachki et al. [53]	continuous operators between Sobolev spaces on the torus, on compact input sets	Fourier neural operator	uniform on compact sets
Lanthaler et al. [65]	continuous operators between function spaces on Euclidean domains, on compact input sets	averaging neural operator (one spatial average plus pointwise nonlinearity)	uniform on compact sets
Gödeke and Fernsel [35]	continuous operators between normed or metric spaces satisfying encoder–decoder approximation properties	encoder–decoder architectures (DeepONet, BasisONet, MIONet, frame based)	cc; also compact-set-independent choice of the architecture
Yao et al. [100] (Thms. 3.2, 4.3)	$C^{1}$ operators between Sobolev spaces on the torus whose derivative is continuous into Hilbert–Schmidt operators	Fourier neural operator with smooth activation	sup over a compact set of the operator error and of the Hilbert–Schmidt derivative error; $L^{2}_{\mu}$ of both for a Borel probability $\mu$ with moment conditions
de Feo [25] (Thms. 4.3, 4.6; Cor. 4.19)	$C^{k}$ operators together with derivatives, on Banach spaces	encoder–decoder classes (DeepONet, Deep-H-ONet, Schauder nets)	not dense in Fréchet $C^{k}$ compact-open or Fréchet–Sobolev norms, already for bounded linear targets; dense in Bastiani $C^{k}$ compact-open (derivatives uniform on compacts, directions from compacts), weighted Bastiani–Sobolev, and Gaussian Sobolev $W^{k,p}_{\gamma}$ norms
This paper	continuous $f\colon\mathcal{H}\to\mathcal{Y}$ , $\mathcal{H}$ separable Hilbert, $\mathcal{Y}$ separable Hilbert	one hidden layer $\sum_{j}v_{j}\sigma(\langle a_{j},x\rangle-b_{j})$ , continuous non-polynomial $\sigma$ of polynomial growth; coefficients from the ridgelet transform of a spectral approximant	cc

Appendix

Table 3: Universal approximation with infinite-dimensional inputs: neural operators, derivatives, and this paper.

Reference	Target class	Approximant	Norm	Bound
Barron [5]	$f$ on a ball in $\mathbb{R}^{d}$ with $\int\|\xi\|\|\widehat{f}(\xi)\|\,\mathrm{d}\xi\leq C$	one hidden layer, sigmoidal, width $N$	$L^{2}(\mu)$ on the ball	$O(C/\sqrt{N})$ , no explicit $d$
Mhaskar and Hahm [72]	smooth functionals on compact subsets of $L^{p}$ or $C([-1,1]^{s})$	generalized translation networks	uniform	order bounds in the number of neurons, with matching lower bounds
Song et al. [85] (Thms. 1, 2)	functionals on $L^{p}([-1,1]^{s})$ with $\omega_{F}(r)\leq cr^{\lambda}$ , on the unit ball $K$ of $C^{\beta}$ (Thm. 2)	functional deep ReLU network: a bounded linear first layer followed by ReLU layers, $M$ nonzero weights, depth $O(L_{M}^{2})$	sup over $K$	$O(L_{M}^{-\beta\lambda/s})$ with $L_{M}=\log M/\log\log M$
Gonon et al. [36] (Thm. 1)	$f$ on a separable Hilbert space with a Fourier–Stieltjes representation, second moments, and a density w.r.t. the sampling measure	random ReLU features, width $N$ , trained readout	expected squared $L^{2}$ error over the input law	$O(N^{-1})$ for the squared error
Korolev [52] (Thm. 3.24)	variation space $F_{1}(X;Y)$ of maps between duals of separable Banach spaces with Schauder bases, Lipschitz for the weak- $*$ metric on $Y$	$\sum_{i=1}^{n}\alpha_{i}(K_{i}x)_{+}$ with finite-rank $K_{i}$ , $\\|K_{i}\\|\leq 1$	weak- $*$ metric on $Y$ , pointwise with factor $\\|x\\|$ ; Bochner $L^{p}_{\mu}$ for $\mu$ with $p$ moments	$n^{-1/2}$ times $c\\|f\\|_{F_{1}}\\|x\\|$ ; inverse theorem
Neufeld and Schmocker [78] (Thm. 4.5; Cor. 4.21)	Barron space $\mathcal{B}^{r}_{G,\theta}(X)$ of a separable Banach space of type $t$ ; in Cor. 4.21, a Barron–ridgelet class on $U\subset\mathbb{R}^{m}$ defined by an admissible filter pair	random feature model with $N$ features; random neural network with $N$ neurons	Bochner $L^{r}$ in the $X$ -norm; weighted Sobolev $W^{k,p}(U,w)$	$N^{-\kappa}$ , $\kappa=1-1/\min(r,t)$ , so $N^{-1/2}$ for type $2$ , $r\geq 2$ ; $\kappa=1-1/\min(2,p,r)$ in Cor. 4.21

Appendix

Table 4: Quantitative upper bounds for functionals and for networks with Hilbert or Banach inputs, as rates in the width

N

Reference	Target class	Approximant	Norm	Bound
Lanthaler et al. [64]	PDE solution operators; general error decomposition	DeepONet	$L^{2}(\mu)$ over the input law	encoder, approximator, reconstruction errors; algebraic rates in examples
Furuya et al. [31] (Thm. 1)	solution operator of a nonlinear parabolic equation on an $L^{\infty}$ ball, short time	weight-tied ReLU neural operator, Green-function expansion of rank $N$	sup over the ball of $L^{r}(0,T;L^{s})$	depth $O((\log\varepsilon^{-1})^{2})$ , width $O(\varepsilon^{-1}(\log\varepsilon^{-1})^{2})$ , $N$ by the Green-function error
Kratsios et al. [58]	uniformly continuous, in particular Lipschitz, operators on compact Sobolev sets	mixture of neural operators routed by a tree	uniform on the compact set	active expert of size $O(\varepsilon^{-1})$ for Lipschitz targets
Adcock et al. [3] (Thm. 4.1)	Lipschitz $F\in L^{2}_{\mu}(X;Y)$ , $\mu$ Gaussian, whose PCA latent representation lies in a Gaussian Sobolev space $H^{k}$	Hermite–PCA approximation from $N_{X},N_{Y}$ PCA samples and $M$ input–output samples	$L^{2}_{\mu}(X;Y)$	PCA tails, sampling terms, and a term improving with the Sobolev order $k$

Appendix

Table 5: Quantitative upper bounds for neural operators and surrogates of operators.

Reference	Target class	Approximant	Norm	Bound
Lanthaler and Stuart [63] (Thm. 2.11, Cor. 2.12)	$r$ -times Fréchet differentiable functionals or operators on a compact $K$ containing an infinite-dimensional hypercube $Q_{\alpha}$	functionals of neural-network type $\Phi\circ L$ , $L$ finite-rank linear, $\Phi$ a ReLU network (covers PCA-Net, DeepONet, NOMAD, FNO)	sup over $K$	complexity $\geq\exp(c\varepsilon^{-\kappa})$ , $\kappa=1/((\alpha+1+\delta)r)$
Lanthaler [62]	Lipschitz operators between Banach spaces	any architecture, size measured in bits	uniform on a compact set; expectation over an input law	size exponential in $\varepsilon^{-1}$ , for every activation
Kovachki et al. [55]	Lipschitz and Fréchet-differentiable operators	any method using $n$ input–output samples ( $n$ -widths)	as stated there	$n$ exponential in $\varepsilon^{-1}$
Adcock et al. [2]	uniformly bounded Lipschitz (or Hölder) operators between separable Hilbert spaces	any estimator from $n$ noisy samples (minimax)	$L^{2}$ risk over an input law with given covariance spectrum	no algebraic rate in $n$ ; sharp for exponentially decaying spectra

Appendix

Table 6: Lower bounds: no algebraic rate exists for the class.

Neural Network Approximation Theory

Neural network approximation theory studies how network size, depth, activations, and structure determine the accuracy of functions and operators.

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Approximating Smooth Functionals with ReLU Networks

Shuhao Jiao

Department of Biostatistics, City University of Hong Kong, Hong Kong SAR, China.

ReLU Neural Networks Neural Network Approximation Theory

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We study the uniform approximation of smooth scalar-valued functionals on an infinite-dimensional separable Hilbert space by ReLU neural networks. A key feature in deep learning for functional data is the varying importance of different coordinates/dimensions. Representing the functional input in a basis expansion, we quantify the importance of each coordinate through both the magnitude of its corresponding basis score and the directional sensitivity of the target functional. Our analysis combines coordinate truncation, anisotropic partitioning, local Taylor approximation, and ReLU network realization, while allowing unrestricted interactions among the retained coordinates. We establish a general nonasymptotic upper bound for the uniform approximation error and a complementary pseudo-dimension-based lower bound for the worst-case approximation error. Under generalized exponential coordinate decay

w_ds_d\asymp\exp(-cd^ρ)

, with

ρ>0

, the upper and lower bounds match at the leading order, which is stretched-exponential in the logarithm of the network size budget, and thus yield the nearly optimal approximation rate. This is the first work to characterize neural network approximation error for infinite-dimensional functional inputs explicitly through the joint dimensional decay of coordinate magnitudes and directional sensitivities.

Abstract

Neural Network Approximation Theory

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