# $k$-means as a variational EM approximation of Gaussian mixture models

## TL;DR

This paper demonstrates that k-means clustering can be derived as a special case of a variational EM approximation for Gaussian Mixture Models, providing new theoretical insights and generalizations.

## Contribution

It shows that k-means arises from truncated variational EM without requiring small Gaussian variances, and links k-means to a free energy framework for better understanding and extensions.

## Key findings

- k-means increases a free energy associated with truncated distributions
- Generalizations of k-means can be derived by considering multiple closest clusters
- Embedding k-means into a free energy framework allows for theoretical interpretation of its variants

## Abstract

We show that $k$-means (Lloyd's algorithm) is obtained as a special case when truncated variational EM approximations are applied to Gaussian Mixture Models (GMM) with isotropic Gaussians. In contrast to the standard way to relate $k$-means and GMMs, the provided derivation shows that it is not required to consider Gaussians with small variances or the limit case of zero variances. There are a number of consequences that directly follow from our approach: (A) $k$-means can be shown to increase a free energy associated with truncated distributions and this free energy can directly be reformulated in terms of the $k$-means objective; (B) $k$-means generalizations can directly be derived by considering the 2nd closest, 3rd closest etc. cluster in addition to just the closest one; and (C) the embedding of $k$-means into a free energy framework allows for theoretical interpretations of other $k$-means generalizations in the literature. In general, truncated variational EM provides a natural and rigorous quantitative link between $k$-means-like clustering and GMM clustering algorithms which may be very relevant for future theoretical and empirical studies.