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

• URL: http://arxiv.org/abs/2605.08559v1
• Date: Fri, 08 May 2026 23:40:30 GMT
• Title: Structure-Preserving Reconstruction of Convex Lipschitz Functionals on Hilbert Spaces from Finite Samples
• Abstract summary: We show that a convex functional on a separable Hilbert space $H$ can be reconstructed up to arbitrary uniform accuracy.<n>We introduce convex neural functionals (CNFs), a structured trainable architecture class containing our reconstruction.<n>CNFs provide a principled foundation for learning convex functionals from finite data.
• Score: 13.6657201205511
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• A Closed-Form Formula for Consistent Lipschitz Regression on Metric Spaces with Sparse Neural Network Realizations [10.292476979020522]
  We introduce a simple closed-form two-stage'' formula $hatf$ for reconstructing an unknown Lipschitz function.<n>Our main result is a high-probability uniform ($Linfty$) recovery guarantee that jointly controls approximation and statistical errors.
  arXiv  Detail & Related papers  (2026-09-02T20:10:29Z)
• Q-Learning with Stable Infinite-Dimensional Linear Function Approximation [10.189658648290257]
  We develop a stable infinite-dimensional linear function approximation framework for Q-learning from a single Markovian behavior-policy trajectory.<n>We illustrate the framework through Q-learning with linear density approximation and output-layer neural weight training under a frozen pretrained network.
  arXiv  Detail & Related papers  (2026-08-23T22:40:11Z)
• Learning and Testing Convex Functions [18.95992615547965]
  We consider the problems of emphlearning and emphtesting real-valued convex functions over Gaussian space.<n>Despite the extensive study of function convexity across mathematics, its learnability and testability have largely been examined only in discrete or restricted settings.
  arXiv  Detail & Related papers  (2025-11-14T17:19:44Z)
• Surrogate to Poincaré inequalities on manifolds for dimension reduction in nonlinear feature spaces [49.1574468325115]
  We aim to approximate a continuously differentiable function $u:mathbbRd rightarrow mathbbRm$ by a composition of functions $fcirc g$ where $g:mathbbRd rightarrow mathbbRm$, $mleq d$, and $f : mathbbRm rightarrow mathbbRR$.<n>For a fixed $g$, we build $f$ using classical regression methods, involving evaluations
  arXiv  Detail & Related papers  (2025-05-03T12:37:27Z)
• Learning linear dynamical systems under convex constraints [4.13951084724473]
  We consider the problem of finite-time identification of linear dynamical systems from samples of a single trajectory.<n>We show that $A*$ can be reliably estimated for values much much smaller than what is needed for the unconstrained setting.
  arXiv  Detail & Related papers  (2023-03-27T11:49:40Z)
• Random matrices in service of ML footprint: ternary random features with no performance loss [55.30329197651178]
  We show that the eigenspectrum of $bf K$ is independent of the distribution of the i.i.d. entries of $bf w$. We propose a novel random technique, called Ternary Random Feature (TRF) The computation of the proposed random features requires no multiplication and a factor of $b$ less bits for storage compared to classical random features.
  arXiv  Detail & Related papers  (2021-10-05T09:33:49Z)
• Submodular + Concave [53.208470310734825]
  It has been well established that first order optimization methods can converge to the maximal objective value of concave functions. In this work, we initiate the determinant of the smooth functions convex body $$F(x) = G(x) +C(x)$. This class of functions is an extension of both concave and continuous DR-submodular functions for which no guarantee is known.
  arXiv  Detail & Related papers  (2021-06-09T01:59:55Z)
• Instance-Dependent Bounds for Zeroth-order Lipschitz Optimization with Error Certificates [0.0]
  We study the problem of zeroth-order (black-box) optimization of a Lipschitz function $f$ defined on a compact subset $mathcal X$ of $mathbb Rd$. We characterize the optimal number of evaluations of any Lipschitz function $f$ to find and certify an approximater of $f$ at accuracy $varepsilon$.
  arXiv  Detail & Related papers  (2021-02-03T09:51:03Z)
• Small Covers for Near-Zero Sets of Polynomials and Learning Latent Variable Models [56.98280399449707]
  We show that there exists an $epsilon$-cover for $S$ of cardinality $M = (k/epsilon)O_d(k1/d)$. Building on our structural result, we obtain significantly improved learning algorithms for several fundamental high-dimensional probabilistic models hidden variables.
  arXiv  Detail & Related papers  (2020-12-14T18:14:08Z)
• Piecewise Linear Regression via a Difference of Convex Functions [50.89452535187813]
  We present a new piecewise linear regression methodology that utilizes fitting a difference of convex functions (DC functions) to the data. We empirically validate the method, showing it to be practically implementable, and to have comparable performance to existing regression/classification methods on real-world datasets.
  arXiv  Detail & Related papers  (2020-07-05T18:58:47Z)
• Convex Regression in Multidimensions: Suboptimality of Least Squares Estimators [4.758195657080579]
  Least Squares Estimators are shown to be suboptimal for estimating a $d$-dimensional convex function in squared error loss when the dimension $d$ is 5 or larger. The specific function classes considered include: (i) bounded convex functions supported on a polytope (in random design), (ii) Lipschitz convex functions supported on any convex domain (in random design), and (iii) convex functions supported on a polytope (in fixed design)
  arXiv  Detail & Related papers  (2020-06-03T04:57:05Z)
• Complexity of Finding Stationary Points of Nonsmooth Nonconvex Functions [84.49087114959872]
  We provide the first non-asymptotic analysis for finding stationary points of nonsmooth, nonsmooth functions. In particular, we study Hadamard semi-differentiable functions, perhaps the largest class of nonsmooth functions.
  arXiv  Detail & Related papers  (2020-02-10T23:23:04Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.