Model-based Clustering using Automatic Differentiation: Confronting Misspecification and High-Dimensional Data

TL;DR

This paper compares gradient descent with automatic differentiation to EM for Gaussian mixture model clustering, addressing misspecification and high-dimensional data challenges, and introduces a new penalized likelihood method.

Contribution

It proposes a novel penalized likelihood approach with AD for improved clustering, especially in high-dimensional settings, and analyzes performance under misspecification.

Findings

GD outperforms EM on high-dimensional data

EM performs better under model misspecification

The penalized likelihood improves cluster interpretation

Abstract

We study two practically important cases of model based clustering using Gaussian Mixture Models: (1) when there is misspecification and (2) on high dimensional data, in the light of recent advances in Gradient Descent (GD) based optimization using Automatic Differentiation (AD). Our simulation studies show that EM has better clustering performance, measured by Adjusted Rand Index, compared to GD in cases of misspecification, whereas on high dimensional data GD outperforms EM. We observe that both with EM and GD there are many solutions with high likelihood but poor cluster interpretation. To address this problem we design a new penalty term for the likelihood based on the Kullback Leibler divergence between pairs of fitted components. Closed form expressions for the gradients of this penalized likelihood are difficult to derive but AD can be done effortlessly, illustrating the advantage of AD-based optimization. Extensions of this penalty for high dimensional data and for model selection are discussed. Numerical experiments on synthetic and real datasets demonstrate the efficacy of clustering using the proposed penalized likelihood approach.