Variational Inference and Sparsity in High-Dimensional Deep Gaussian Mixture Models

TL;DR

This paper introduces a scalable variational inference method for deep Gaussian mixture models with sparsity priors, enhancing clustering in high-dimensional data by automatically regularizing model complexity.

Contribution

It develops a natural gradient variational inference algorithm for deep mixtures of factor analyzers with sparsity priors, enabling efficient high-dimensional clustering.

Findings

Sparsity priors improve clustering accuracy.

Overfitted mixtures allow automatic component selection.

Method performs well on simulated and real high-dimensional data.

Abstract

Gaussian mixture models are a popular tool for model-based clustering, and mixtures of factor analyzers are Gaussian mixture models having parsimonious factor covariance structure for mixture components. There are several recent extensions of mixture of factor analyzers to deep mixtures, where the Gaussian model for the latent factors is replaced by a mixture of factor analyzers. This construction can be iterated to obtain a model with many layers. These deep models are challenging to fit, and we consider Bayesian inference using sparsity priors to further regularize the estimation. A scalable natural gradient variational inference algorithm is developed for fitting the model, and we suggest computationally efficient approaches to the architecture choice using overfitted mixtures where unnecessary components drop out in the estimation. In a number of simulated and two real examples, we demonstrate the versatility of our approach for high-dimensional problems, and demonstrate that the use of sparsity inducing priors can be helpful for obtaining improved clustering results.