Quantitative Stability and Numerical Resolution of the Moment Measure Problem
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917400219222016 |
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| author | Bonnet, Guillaume Rubinstein, Yanir A. |
| author_facet | Bonnet, Guillaume Rubinstein, Yanir A. |
| contents | The moment measure problem consists in finding a convex function $ψ$ whose moment measure, i.e., the pushforward by $\nabla ψ$ of the measure with density $e^{-ψ(\,\cdot\,)}$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_09914 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative Stability and Numerical Resolution of the Moment Measure Problem Bonnet, Guillaume Rubinstein, Yanir A. Functional Analysis Numerical Analysis The moment measure problem consists in finding a convex function $ψ$ whose moment measure, i.e., the pushforward by $\nabla ψ$ of the measure with density $e^{-ψ(\,\cdot\,)}$, 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. |
| title | Quantitative Stability and Numerical Resolution of the Moment Measure Problem |
| topic | Functional Analysis Numerical Analysis |
| url | https://arxiv.org/abs/2604.09914 |