Bayesian Spectral Clustering

Abstract: We introduce Bayesian Spectral Clustering (BSC), a probabilistic reformulation of spectral clustering. Classical spectral clustering relies on a hand-crafted affinity graph (e.g., Gaussian kernel with $k$-NN sparsification) that is then treated as fixed, and recent improvements typically optimize that graph jointly with the clustering objective. However, these approaches still output a single graph and a single hard partition, providing neither principled quantification of uncertainty nor a guaranteed notion of when the learned affinities are reliable. BSC addresses this by treating the affinity matrix $W$ itself as a latent variable with sparsity- and locality-promoting priors, linking $W$ to the observed data through a Laplacian-smoothness likelihood, and performing variational inference to obtain a joint posterior over $W$ and the cluster assignments. We prove that (i) the standard Gaussian affinity emerges as the maximum a posteriori edge weight, giving a probabilistic justification for the classical kernel, and (ii) the posterior-mean graph is the unique minimizer of a strictly convex objective and is automatically sparse. Empirically, BSC attains state-of-the-art clustering quality while producing calibrated per-sample assignment confidence.

Submission Type: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Uri_Shaham1

Submission Number: 7014