Sparse Mixture Models inspired by ANOVA Decompositions

TL;DR

This paper introduces a sparse mixture modeling approach inspired by ANOVA decompositions, using Gaussian and von Mises distributions on the high-dimensional torus to efficiently approximate functions with limited variable interactions.

Contribution

It proposes a novel sparse mixture model framework with regularized EM algorithm and statistical tests for variable interaction analysis on the torus.

Findings

Effective approximation of functions with limited variable interactions.

Successful application of regularized EM for sparse mixture learning.

Numerical examples demonstrate the approach's performance.

Abstract

Inspired by the analysis of variance (ANOVA) decomposition of functions we propose a Gaussian-Uniform mixture model on the high-dimensional torus which relies on the assumption that the function we wish to approximate can be well explained by limited variable interactions. 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 examples demonstrate the performance of our approach.