Sparse ANOVA Inspired Mixture Models

Based on the analysis of variance (ANOVA) decomposition of functions which relies on the assumption that the function we wish to approximate can be well explained by limited variable interaction, we propose a sparse Gaussian-like mixture model on the high dimensional torus. We consider three approaches, namely wrapped Gaussians, diagonal wrapped Gaussians and products of von Mises distributions. The sparsity of the mixture model is ensured by the fact that its summands are products of Gaussian-like density functions acting on low dimensional spaces and uniform probability densities defined on the remaining directions. To learn such a sparse mixture model from given samples, we propose an objective function consisting of the negative log-likelihood function of the mixture model and a regularizer that penalizes the number of its summands. For minimizing this functional we combine the Expectation Maximization algorithm with a proximal step that takes the regularizer into account. To decide which summands of the mixture model are important, we apply a Kolmogorov-Smirnov test. Numerical example demonstrate the performance of our approach.