Mathematics > Optimization and Control
arXiv:2211.01804 (math)
[Submitted on 2 Nov 2022 (v1), last revised 4 Oct 2023 (this version, v5)]
Title:Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels
View a PDF of the paper titled Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels, by Johannes Hertrich and 3 other authors
View PDFAbstract:The aim of this paper is twofold. Based on the geometric Wasserstein tangent space, we first introduce Wasserstein steepest descent flows. These are locally absolutely continuous curves in the Wasserstein space whose tangent vectors point into a steepest descent direction of a given functional. This allows the use of Euler forward schemes instead of Jordan--Kinderlehrer--Otto schemes. For $\lambda$-convex functionals, we show that Wasserstein steepest descent flows are an equivalent characterization of Wasserstein gradient flows. The second aim is to study Wasserstein flows of the maximum mean discrepancy with respect to certain Riesz kernels. The crucial part is hereby the treatment of the interaction energy. Although it is not $\lambda$-convex along generalized geodesics, we give analytic expressions for Wasserstein steepest descent flows of the interaction energy starting at Dirac measures. In contrast to smooth kernels, the particle may explode, i.e., a Dirac measure becomes a non-Dirac one. The computation of steepest descent flows amounts to finding equilibrium measures with external fields, which nicely links Wasserstein flows of interaction energies with potential theory. Finally, we provide numerical simulations of Wasserstein steepest descent flows of discrepancies.
| Subjects: | Optimization and Control (math.OC); Numerical Analysis (math.NA); Probability (math.PR) |
| Cite as: | arXiv:2211.01804 [math.OC] |
| (orarXiv:2211.01804v5 [math.OC] for this version) | |
| https://doi.org/10.48550/arXiv.2211.01804 arXiv-issued DOI via DataCite | |
| Journal reference: | Journal of Mathematical Analysis and Applications, vol. 531, 127829, 2024 |
| Related DOI: | https://doi.org/10.1016/j.jmaa.2023.127829 DOI(s) linking to related resources |
Submission history
From: Johannes Hertrich [view email][v1] Wed, 2 Nov 2022 16:10:28 UTC (6,036 KB)
[v2] Fri, 11 Nov 2022 10:02:52 UTC (6,037 KB)
[v3] Mon, 16 Jan 2023 09:39:10 UTC (6,014 KB)
[v4] Wed, 22 Mar 2023 08:47:29 UTC (6,019 KB)
[v5] Wed, 4 Oct 2023 07:45:35 UTC (4,324 KB)
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View a PDF of the paper titled Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels, by Johannes Hertrich and 3 other authors
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