Clustering Mixtures of Discrete Distributions: A Note on Mitra's Algorithm
Mohamed Seif, Yanxi Chen
TL;DR
This note sharpens the theoretical guarantees for Mitra's algorithm, a two-step spectral clustering method for mixtures of discrete distributions, by adapting the analysis to bipartite stochastic block models. Under precise spectral and separation conditions, including a necessary lower bound on $m\sigma^2$, the Centers and Assignment lemmas jointly yield exact clustering with high probability. The general-case result is then specialized to the B-SBM, deriving a language of separation in terms of $(p-q)^2/\sigma^2$ and the right-vertex distinctness $\Delta_V$, with a quantitative corollary governed by contrast parameters and graph size. Overall, the work provides refined, likelihood-based recovery guarantees for Mitra's approach in both general and bipartite discrete-miffusion settings, highlighting conditions under which tiny clusters can be reliably detected.
Abstract
In this note, we provide a refined analysis of Mitra's algorithm \cite{mitra2008clustering} for classifying general discrete mixture distribution models. Built upon spectral clustering \cite{mcsherry2001spectral}, this algorithm offers compelling conditions for probability distributions. We enhance this analysis by tailoring the model to bipartite stochastic block models, resulting in more refined conditions. Compared to those derived in \cite{mitra2008clustering}, our improved separation conditions are obtained.