Spectral Clustering, Bayesian Spanning Forest, and Forest Process

TL;DR

This paper introduces a Bayesian forest model that enhances spectral clustering by providing uncertainty quantification and flexible model extensions, demonstrated through superior performance and applications in high-dimensional and multi-view data.

Contribution

The paper develops a Bayesian forest model with a forest process extension, linking it to spectral clustering and enabling uncertainty quantification and flexible data modeling.

Findings

Posterior connecting matrix in the forest model closely matches spectral clustering eigenvectors.

The proposed MCMC algorithm outperforms existing spectral clustering methods.

Model extensions improve clustering in high-dimensional and multi-view data.

Abstract

Spectral clustering views the similarity matrix as a weighted graph, and partitions the data by minimizing a graph-cut loss. Since it minimizes the across-cluster similarity, there is no need to model the distribution within each cluster. As a result, one reduces the chance of model misspecification, which is often a risk in mixture model-based clustering. Nevertheless, compared to the latter, spectral clustering has no direct ways of quantifying the clustering uncertainty (such as the assignment probability), or allowing easy model extensions for complicated data applications. To fill this gap, we propose the Bayesian forest model as a generative graphical model for spectral clustering. This is motivated by our discovery that the posterior connecting matrix in a forest model has almost the same leading eigenvectors, as the ones used by normalized spectral clustering. To induce a distribution for the forest, we develop a ``forest process'' as a graph extension to the urn process, while we carefully characterize the differences in the partition probability. We derive a simple Markov chain Monte Carlo algorithm for posterior estimation, and demonstrate superior performance compared to existing algorithms. We illustrate several model-based extensions useful for data applications, including high-dimensional and multi-view clustering for images.