Gaussian Mixture Models

K-Means is Low-Variance Limit of Expectation Maximization in GMM

The k-means clustering problem can be derived from a zero-variance limit of expectation maximization in a Gaussian mixture model. For a finite GMM with \(N\) observations \(\{x_n\}\), we want to infer cluster assignments \(\{z_n\}\) and each cluster’s Gaussian parameters Gaussian parameters \(\{\mu_c, \Sigma_c\}\). Let \(\pi_c\) denote the mixing proportion of the \(c\)th class. The marginal distribution of an observation is:

\[p(x_n) = \sum_{c=1}^C p(x_n | z_n = c) p(z_n = c) = \sum_{c=1}^C \mathcal{N}(x_n; \mu_c, \Sigma_c) \pi_c\]

The cluster assignments and cluster parameters can be learnt using Expectation Maximization. Assuming the clusters each have equal covariance \(\sigma I\), the E step takes the following form:

\[\begin{align*} p(z_n = c | x_n) &= \frac{p(x_n | z_n = c) p(z_n = c)}{p(x_n)}\\ &= \frac{\exp(-\frac{1}{2\sigma} ||x_n - \mu_c||_2^2) \pi_c}{\sum_c \exp(-\frac{1}{2\sigma} ||x_n - \mu_c||_2^2) \pi_c} \end{align*}\]

As \(\sigma \rightarrow 0\), all mass is placed on the nearest centroid, just as we do in k-means clustering. The M step then updates the centroids (cluster means) to the average of the points assigned to that particular cluster:

\[\mu_c = \frac{1}{|X_c|} \sum_{x_n \in X_c} x_n\]

where \(X_c = \{ x_n \colon p(z_n = c \lvert x_n) = 1 \}\) is the set of observations assigned to the \(c\)th cluster.

We can phrase this as an optimization problem with the objective function:

\[\min_{\{\mu_c\}, \{z_n\}} \sum_{n=1}^N \mathbb{I}(z_n = c) ||x_n - \mu_c||_2^2\]

where the cluster centroids are defined above.