Structure-Preserving Reconstruction of Convex Lipschitz Functionals on Hilbert Spaces from Finite Samples
For researchers in optimization, risk analysis, and machine learning who need to learn convex functionals from data, this provides a principled reconstruction method with theoretical guarantees, though the result is theoretical and incremental in nature.
The paper proves that any convex Lipschitz functional on a compact convex subset of a separable Hilbert space can be uniformly approximated to arbitrary accuracy by a finite-sample reconstruction that preserves convexity and Lipschitz regularity, using only finitely many linear measurements. The reconstruction is implementable by a ReLU-MLP, and the authors introduce convex neural functionals (CNFs) as a trainable architecture class with guaranteed convexity and Lipschitzness.
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.