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

2026-05-08

AI Summary

This work addresses the problem of accurately reconstructing convex, L-Lipschitz functionals defined on a Hilbert space from a finite number of linear measurements, while rigorously preserving both convexity and the Lipschitz constant. The authors introduce the first explicit, finitely computable reconstruction formula and propose a novel Convex Neural Functional (CNF) architecture—a structured, trainable model class based on ReLU multilayer perceptrons. Their approach achieves uniform ε-accuracy in functional reconstruction over any compact convex set, offering strong theoretical guarantees alongside practical implementability. This framework establishes a new paradigm for learning convex functionals from limited data, effectively balancing structural constraints with expressive power.

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 $ρ:C\to\mathbb{R}$, 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.

Problem

• convex functionals
• Lipschitz continuity
• finite samples
• Hilbert spaces
• structure-preserving reconstruction

Innovation

• convex neural functionals
• structure-preserving reconstruction
• Lipschitz convex functionals
• finite-sample learning
• ReLU-MLP representation