Rylan Schaeffer

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25 February 2021

Revisiting k-means - New Algorithms via Bayesian Nonparametrics

by Kulis and Jordan (ICML 2012)

Motivation

One limitation of k-means clustering is that the number of clusters must be specified in advance. Here, Kulis and Jordan show that a low-variance asymptotic analysis of a Dirichlet process Gaussian Mixture Model (DP-GMM) yields a k-means-like clustering algorithm that adds new clusters when justified by the data.

K-Means as Low-Variance Approximation of Expectation Maximization in Gaussian Mixture Model

We quickly review k-means clustering and how it can be derived from a low variance asymptotic approximation of expectation maximization applied to 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.

Low-Variance Approximation of Dirichlet Process Gaussian Mixture Model

In a DP-GMM, we now have an infinite number of clusters, enabled through the Dirichlet Process. Specifically, we draw an infinite number of cluster centroids \(\{\mu_c \}_{c=1}^{\infty}\) from a base distribution \(G_0\) and draw mixing proportions \(\{\pi_c\}_{c=1}^{\infty}\) from \(DP(\alpha, G_0)\). Then, for each observation, we posit that we draw

\[\begin{align*} \mu_1, \mu_2, ... &\sim G_0 \end{align*}\] tags: dirichlet-process - k-means - clustering - bayesian-nonparametrics - mixture-models