On the best approximation by finite Gaussian mixtures

We consider the problem of approximating a general Gaussian location mixture by finite mixtures. The minimum order of finite mixtures that achieve a prescribed accuracy (measured by various f-divergences) are determined within constant factors for the family of compactly supported or subgaussian mixing distributions. While the upper bound is achieved using the technique of local moment matching, the lower bound is established by relating the best approximation error to the low-rank approximation of certain trigonometric moment matrices and weighted moment matrices, followed by a refined spectral analysis of the minimum eigenvalue of these matrices. In the case of Gaussian mixing distributions, this result corrects a previous lower bound in [1].