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 « \m,\bSigma » of the Gaussians is supported on a given compact semi-algebraic set « S ». 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 by using a well-known specific property of Gaussian measures, this optimal transport is then viewed as a Generalized Moment Problem (GMP) for which we provide a ``mesh-free" numerical scheme. In particular, we do not assume that the mixing measure is finitely supported nor that the variance is the same for all components.
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