Hard-Clustering with Gaussian Mixture Models

TL;DR

This paper introduces two algorithms for a restricted version of the Gaussian mixture model clustering problem, providing approximate solutions with guarantees on their quality, addressing limitations of existing methods like CEM.

Contribution

The paper proposes novel algorithms for a restricted CMLE problem that achieve near-optimal solutions with provable approximation guarantees.

Findings

Algorithms compute solutions within a factor (1+ε) of optimal

Addresses issues with standard CEM convergence

Provides solutions satisfying natural properties of the CMLE problem

Abstract

Training the parameters of statistical models to describe a given data set is a central task in the field of data mining and machine learning. A very popular and powerful way of parameter estimation is the method of maximum likelihood estimation (MLE). Among the most widely used families of statistical models are mixture models, especially, mixtures of Gaussian distributions. A popular hard-clustering variant of the MLE problem is the so-called complete-data maximum likelihood estimation (CMLE) method. The standard approach to solve the CMLE problem is the Classification-Expectation-Maximization (CEM) algorithm. Unfortunately, it is only guaranteed that the algorithm converges to some (possibly arbitrarily poor) stationary point of the objective function. In this paper, we present two algorithms for a restricted version of the CMLE problem. That is, our algorithms approximate reasonable solutions to the CMLE problem which satisfy certain natural properties. Moreover, they compute solutions whose cost (i.e. complete-data log-likelihood values) are at most a factor $(1+\epsilon)$ worse than the cost of the solutions that we search for. Note the CMLE problem in its most general, i.e. unrestricted, form is not well defined and allows for trivial optimal solutions that can be thought of as degenerated solutions.