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

arXiv: https://arxiv.org/abs/2605.08559 · HTML: https://mlsanity.com/paper/2605.08559/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 55 references that are in the map.
- First posted on arXiv 8 May 2026.

## Builds on (most cited first, 21 of 21)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
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- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
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- [Universal Approximation Theorems for Differentiable Geometric Deep Learning](https://mlsanity.com/paper/2101.05390/) (2021)
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- [Mathematical theory of deep learning](https://mlsanity.com/paper/2407.18384/) (2024)
- [A Computational Framework for Multivariate Convex Regression and its Variants](https://mlsanity.com/paper/1509.08165/) (2015)
- [Mixtures of Neural Operators Reduce Active Complexity in Operator Learning](https://mlsanity.com/paper/2404.09101/) (2024)
- [Deep calibration of rough stochastic volatility models](https://mlsanity.com/paper/1810.03399/) (2018)
- [A three layer neural network can represent any multivariate function](https://mlsanity.com/paper/2012.03016/) (2020)
- [Bridging the Gap Between Approximation and Learning via Optimal Approximation by ReLU MLPs of Maximal Regularity](https://mlsanity.com/paper/2409.12335/) (2024)
- [Optimal Nonparametric Inference via Deep Neural Network](https://mlsanity.com/paper/1902.01687/) (2019)
- [Do ReLU Networks Have An Edge When Approximating Compactly-Supported Functions?](https://mlsanity.com/paper/2204.11231/) (2022)
- [Generating Rectifiable Measures through Neural Networks](https://mlsanity.com/paper/2412.05109/) (2024)
- [Max-affine regression via first-order methods](https://mlsanity.com/paper/2308.08070/) (2023)

## Abstract (arXiv)

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 Csubseteq H, every L-Lipschitz convex functional ρ:C→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,·rangleH, with b lying in a finite-dimensional subspace of H, and is exactly implementable by a operatornameReLU-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.