$k$-means as a variational EM approximation of Gaussian mixture models
This provides a rigorous theoretical link between k-means and GMMs, which is incremental but relevant for future studies in clustering algorithms.
The paper tackles the problem of relating k-means clustering to Gaussian mixture models by showing that k-means is a special case of truncated variational EM approximations applied to GMMs with isotropic Gaussians, without requiring small or zero variances. This result leads to consequences such as demonstrating that k-means increases a free energy linked to the k-means objective and enabling generalizations by considering multiple closest clusters.
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.