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arxiv: 2604.09914 · v1 · submitted 2026-04-10 · 🧮 math.FA · cs.NA· math.NA

Quantitative Stability and Numerical Resolution of the Moment Measure Problem

Guillaume Bonnet , Yanir A. Rubinstein This is my paper

Pith reviewed 2026-05-10 16:19 UTC · model grok-4.3

classification 🧮 math.FA cs.NAmath.NA
keywords measureproblemmomentestimatenumericalstabilityoptimalprescribed
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The pith

A quantitative stability estimate for the moment measure problem enables its numerical solution via discrete approximations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a quantitative stability estimate showing that small changes in the prescribed moment measure correspond to small changes in the underlying convex function. This result both confirms the robustness of solutions and motivates a practical numerical method: approximate the target measure by one with finite support, then solve the resulting finite-dimensional problem using Newton's method. Numerical tests reveal that the approximations converge at rates faster than those guaranteed by the stability bound alone. A sympathetic reader would value this because the moment measure problem is highly nonlinear and previously lacked reliable computational tools.

Core claim

We establish a quantitative stability estimate for the moment measure problem. This estimate validates, as well as leads us to introduce, an approach to the numerical resolution of the moment measure problem inspired by semi-discrete optimal transport, consisting in approximating the prescribed measure by a finitely supported one. We describe a Newton method for solving the discrete problem thus obtained, and perform numerical experiments, studying the experimental rates of convergence of the approximation beyond the predictions of the stability estimate.

What carries the argument

The quantitative stability estimate, which provides a bound on the difference between two convex functions in terms of the difference between their associated moment measures.

Load-bearing premise

The existence and uniqueness of a convex solution to the continuous moment measure problem for the given regularity of the target measure.

What would settle it

A counterexample where the discrete approximations fail to converge to the same limit as the support size increases, or where stability fails for some smooth measures.

Figures

Figures reproduced from arXiv: 2604.09914 by Guillaume Bonnet, Yanir A. Rubinstein.

Figure 1
Figure 1. Figure 1: Approximation error, with respect to the number [PITH_FULL_IMAGE:figures/full_fig_p025_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Residuals ∥∇Eν(Φ(k) )∥/∥(ν({yi}))1≤i≤N ∥ in Algorithm 4.1, with respect to the iteration number k, for different values of the discretization parameter n. The dotted line corresponds to the tolerance ε = 10−10. Iterations k at which damping occurs, i.e., for which τ (k) = 1 ̸ , are indicated by round marks. t−n/2−1 < t−n/2 and tn/2+1 > tn (whose exact value will have no effect on ν). We let ν := nX2 j1=− n… view at source ↗
Figure 3
Figure 3. Figure 3: Points of supp ν in test case 5, for some values of the discretization parameter n. cases for which the exact solution is unknown. Rather, the numerical results indicate that it may be of interest in future work to study the possibility of choosing ν adaptively without knowledge of the exact solution. Error norms. The error ∥e −ψµ( · ) − e −ψν ( · )∥L1(Rd) from Theorem 1.3 is not easily computable. Rather,… view at source ↗
read the original abstract

The moment measure problem consists in finding a convex function $\psi$ whose moment measure, i.e., the pushforward by $\nabla \psi$ of the measure with density $e^{-\psi(\,\cdot\,)}$, is prescribed. It is highly non-linear and less understood than the related optimal transport problem. We establish a quantitative stability estimate for this problem. This estimate validates, as well as leads us to introduce, an approach to the numerical resolution of the moment measure problem inspired by semi-discrete optimal transport, consisting in approximating the prescribed measure by a finitely supported one. We describe a Newton method for solving the discrete problem thus obtained, and perform numerical experiments, studying the experimental rates of convergence of the approximation beyond the predictions of the stability estimate.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript establishes a quantitative stability estimate for the moment measure problem (finding a convex function whose moment measure matches a prescribed one) under suitable regularity assumptions on the target measure. This estimate is used to justify and develop a numerical scheme that approximates the prescribed measure by a finitely supported discrete measure, solves the resulting finite-dimensional problem via a Newton method inspired by semi-discrete optimal transport, and reports numerical experiments that examine observed convergence rates, which appear to exceed the theoretical predictions from the stability estimate.

Significance. If the stability estimate is valid, the work provides a rigorous foundation for controlling approximation errors in a nonlinear problem that is less developed than optimal transport. The numerical approach offers a practical resolution method, and the experimental convergence study supplies empirical insight beyond the theory. Strengths include the direct link from stability to discretization and the reproducible numerical protocol implied by the experiments.

major comments (1)
  1. The central stability estimate and its application to discretization rest on the existence and uniqueness of a convex solution to the continuous problem. The manuscript should explicitly state the precise regularity hypotheses on the target measure (e.g., in the introduction or §2) and either prove or cite a reference establishing these properties under exactly those hypotheses; without this, the quantitative bound cannot be applied unconditionally.
minor comments (3)
  1. Notation for the moment measure (pushforward by ∇ψ of e^{-ψ} dx) should be introduced once with a clear equation number and used consistently thereafter.
  2. The description of the Newton solver for the discrete problem would benefit from an explicit statement of the Jacobian or linear system solved at each iteration, perhaps in §4.
  3. Figure captions for the numerical experiments should include the specific mesh sizes, tolerance values, and number of trials used to compute the observed rates.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the positive overall assessment, and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: The central stability estimate and its application to discretization rest on the existence and uniqueness of a convex solution to the continuous problem. The manuscript should explicitly state the precise regularity hypotheses on the target measure (e.g., in the introduction or §2) and either prove or cite a reference establishing these properties under exactly those hypotheses; without this, the quantitative bound cannot be applied unconditionally.

    Authors: We thank the referee for this observation. The stability estimate is indeed conditional on the existence and uniqueness of a convex solution, which we establish under the regularity assumptions placed on the target measure (detailed in Section 2). We agree that these hypotheses should be stated more prominently to make the scope of the result fully transparent. In the revised manuscript we will add an explicit summary of the precise assumptions already in the introduction and include a citation to the reference establishing existence and uniqueness under exactly those hypotheses. This clarification will ensure the quantitative bound is applied only where the underlying well-posedness holds. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes a quantitative stability estimate for the moment measure problem as an independent result under stated regularity assumptions on the target measure. This estimate is then used to motivate and validate a numerical approximation by finitely supported measures, followed by a Newton solver and experimental convergence studies. No load-bearing step reduces by construction to its inputs, no self-definitional relations appear, and no self-citation chains are invoked to force uniqueness or ansatzes. The stability result supplies genuine external support for the numerical method rather than being derived from it, rendering the overall chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all technical assumptions are implicit in the existence of solutions to the moment measure problem.

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