Structure-Preserving Reconstruction of Convex Lipschitz Functionals on Hilbert Spaces from Finite Samples

TL;DR

This paper presents a method to reconstruct convex Lipschitz functionals on Hilbert spaces from finite samples, ensuring the reconstruction preserves convexity and Lipschitz properties, and can be implemented by neural networks.

Contribution

It introduces an explicit finite-sample reconstruction method for convex functionals that maintains convexity and Lipschitz regularity, and develops convex neural functionals for learning from data.

Findings

Reconstruction is convex, Lipschitz, and uniformly accurate from finite samples.

Construction uses finitely many linear measurements in a finite-dimensional subspace.

Reconstruction is implementable by a ReLU-MLP.

Abstract

Convex functionals are ubiquitous in applied analysis, appearing as value functions, risk measures, super-hedging prices, and loss functionals in machine learning. In many applications, however, the functional is only observed through finitely many exact pointwise evaluations. We ask whether a convex functional on a separable Hilbert space $H$ can be reconstructed, up to arbitrary uniform accuracy, by an explicit formula which preserves convexity and Lipschitz regularity and is finitely computable. We answer this affirmatively. For every compact convex $C\subseteq H$, every $L$-Lipschitz convex functional $\rho:C\to\mathbb{R}$, and every $\varepsilon>0$, we construct an explicit finite-sample reconstruction which is convex, $L$-Lipschitz, and uniformly $\varepsilon$-accurate on $C$. The construction uses only finitely many linear measurements $\langle b,\cdot\rangle_H$, with $b$ lying in a finite-dimensional subspace of $H$, and is exactly implementable by a $\operatorname{ReLU}$-MLP. Building on this, we introduce convex neural functionals (CNFs), a structured trainable architecture class containing our reconstruction, whose every admissible parameter configuration is automatically convex and Lipschitz, providing a principled foundation for learning convex functionals from finite data.