Gaussian mixtures closest to a given measure via optimal transport
Given a determinate (multivariate) probability measure $\mu $, we characterize Gaussian mixtures $\nu _\phi $ which minimize the Wasserstein distance $W_2(\mu ,\nu _\phi )$ to $\mu $ when the mixing probability measure $\phi $ on the parameters $(\mathbf{m},\mathbf{\Sigma })$ of the Gaussians is sup...
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.657/ |
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| author | Lasserre, Jean B. |
| author_facet | Lasserre, Jean B. |
| author_sort | Lasserre, Jean B. |
| collection | DOAJ |
| description | Given a determinate (multivariate) probability measure $\mu $, we characterize Gaussian mixtures $\nu _\phi $ which minimize the Wasserstein distance $W_2(\mu ,\nu _\phi )$ to $\mu $ when the mixing probability measure $\phi $ on the parameters $(\mathbf{m},\mathbf{\Sigma })$ of the Gaussians is supported on a compact set $S$. (i) We first show that such mixtures are optimal solutions of a particular optimal transport (OT) problem where the marginal $\nu _{\phi }$ of the OT problem is also unknown via the mixing measure variable $\phi $. Next (ii) by using a well-known specific property of Gaussian measures, this optimal transport is then viewed as a Generalized Moment Problem (GMP) and if the set $S$ of mixture parameters $(\mathbf{m},\mathbf{\Sigma })$ is a basic compact semi-algebraic set, we provide a “mesh-free” numerical scheme to approximate as closely as desired the optimal distance by solving a hierarchy of semidefinite relaxations of increasing size. In particular, we neither assume that the mixing measure is finitely supported nor that the variance is the same for all components. If the original measure $\mu $ is not a Gaussian mixture with parameters $(\mathbf{m},\mathbf{\Sigma })\in S$, then a strictly positive distance is detected at a finite step of the hierarchy. If the original measure $\mu $ is a Gaussian mixture with parameters $(\mathbf{m},\mathbf{\Sigma })\in S$, then all semidefinite relaxations of the hierarchy have same zero optimal value. Moreover if the mixing measure is atomic with finite support, its components can sometimes be extracted from an optimal solution at some semidefinite relaxation of the hierarchy when Curto & Fialkow’s flatness condition holds for some moment matrix. |
| format | Article |
| id | doaj-art-9885b8d64ca44b09856de14e5627b22e |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-9885b8d64ca44b09856de14e5627b22e2025-02-07T11:23:50ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G111455147310.5802/crmath.65710.5802/crmath.657Gaussian mixtures closest to a given measure via optimal transportLasserre, Jean B.0https://orcid.org/0000-0003-0860-9913LAAS-CNRS and Toulouse School of Economics (TSE), BP 54200, 7 Avenue du Colonel Roche, 31031 Toulouse cédex 4, FranceGiven a determinate (multivariate) probability measure $\mu $, we characterize Gaussian mixtures $\nu _\phi $ which minimize the Wasserstein distance $W_2(\mu ,\nu _\phi )$ to $\mu $ when the mixing probability measure $\phi $ on the parameters $(\mathbf{m},\mathbf{\Sigma })$ of the Gaussians is supported on a compact set $S$. (i) We first show that such mixtures are optimal solutions of a particular optimal transport (OT) problem where the marginal $\nu _{\phi }$ of the OT problem is also unknown via the mixing measure variable $\phi $. Next (ii) by using a well-known specific property of Gaussian measures, this optimal transport is then viewed as a Generalized Moment Problem (GMP) and if the set $S$ of mixture parameters $(\mathbf{m},\mathbf{\Sigma })$ is a basic compact semi-algebraic set, we provide a “mesh-free” numerical scheme to approximate as closely as desired the optimal distance by solving a hierarchy of semidefinite relaxations of increasing size. In particular, we neither assume that the mixing measure is finitely supported nor that the variance is the same for all components. If the original measure $\mu $ is not a Gaussian mixture with parameters $(\mathbf{m},\mathbf{\Sigma })\in S$, then a strictly positive distance is detected at a finite step of the hierarchy. If the original measure $\mu $ is a Gaussian mixture with parameters $(\mathbf{m},\mathbf{\Sigma })\in S$, then all semidefinite relaxations of the hierarchy have same zero optimal value. Moreover if the mixing measure is atomic with finite support, its components can sometimes be extracted from an optimal solution at some semidefinite relaxation of the hierarchy when Curto & Fialkow’s flatness condition holds for some moment matrix.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.657/Gaussian mixturesWasserstein distancesemidefinite relaxationsMoment-SOS hierarchy |
| spellingShingle | Lasserre, Jean B. Gaussian mixtures closest to a given measure via optimal transport Comptes Rendus. Mathématique Gaussian mixtures Wasserstein distance semidefinite relaxations Moment-SOS hierarchy |
| title | Gaussian mixtures closest to a given measure via optimal transport |
| title_full | Gaussian mixtures closest to a given measure via optimal transport |
| title_fullStr | Gaussian mixtures closest to a given measure via optimal transport |
| title_full_unstemmed | Gaussian mixtures closest to a given measure via optimal transport |
| title_short | Gaussian mixtures closest to a given measure via optimal transport |
| title_sort | gaussian mixtures closest to a given measure via optimal transport |
| topic | Gaussian mixtures Wasserstein distance semidefinite relaxations Moment-SOS hierarchy |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.657/ |
| work_keys_str_mv | AT lasserrejeanb gaussianmixturesclosesttoagivenmeasureviaoptimaltransport |