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 $(m,\Sigma)$ 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 $(m,\Sigma)$ 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 $(m,\Sigma)\in S$, 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 $(m,\Sigma)\in S$, 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.

翻译：给定一个确定（多元）概率测度 $\mu$，我们刻画了当混合参数 $(m,\Sigma)$（对应高斯的均值与协方差）上的混合概率测度 $\phi$ 支撑于紧集 $S$ 时，最小化与 $\mu$ 之间 Wasserstein 距离 $W_2(\mu,\nu_\phi)$ 的高斯混合分布 $\nu_\phi$。（i）我们首先证明，这类混合分布是某一特殊最优传输问题的最优解，其中该最优传输问题的边际分布 $\nu_\phi$ 通过混合测度变量 $\phi$ 而未知。接着（ii）利用高斯测度一个已知的特殊性质，该最优传输问题被转化为广义矩问题（GMP）。若混合参数集 $S$ 是紧的基本半代数集，我们提出一种无网格数值方案，通过求解一系列尺寸递增的半定松弛问题，对最优距离进行任意精度的逼近。特别地，我们既不假设混合测度具有有限支撑，也不假设所有分量具有相同方差。若原始测度 $\mu$ 本身并非参数属于 $S$ 的高斯混合分布，则该层次结构会在有限步后检测到严格正的距离。若原始测度 $\mu$ 是参数属于 $S$ 的高斯混合分布，则层次结构中所有半定松弛问题的最优解均为零。此外，当混合测度具有有限支撑的原子分布时，若 Curto 与 Fialkow 的平坦条件对某一矩矩阵成立，则有时可从该层次结构的半定松弛问题的最优解中提取其分量。