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.

翻译：凸泛函在应用分析中无处不在，表现为价值函数、风险度量、超对冲价格以及机器学习中的损失泛函。然而在许多应用中，泛函仅能通过有限个精确点值观测获得。我们探究：在可分Hilbert空间$H$上，能否通过显式公式——该公式需保持凸性与Lipschitz正则性且可有限计算——实现凸泛函到任意一致精度的重构。对此我们给出肯定回答。对于每个紧凸集$C\subseteq H$、每个$L$-Lipschitz凸泛函$ρ:C\to\mathbb{R}$及任意$\varepsilon>0$，我们构造了显式的有限样本重构，该重构是凸的、$L$-Lipschitz的，且在$C$上达到$\varepsilon$一致精度。该构造仅使用有限个线性测量$\langle b,\cdot\rangle_H$（其中$b$位于$H$的有限维子空间内），并由$\operatorname{ReLU}$-MLP精确实现。在此基础上，我们引入凸神经泛函（CNF）——一种包含上述重构的结构化可训练架构类，其每个可容许参数配置自动满足凸性与Lipschitz条件，为从有限数据学习凸泛函提供了原理性基础。