Minimax Optimal Clustering of Bipartite Graphs with a Generalized Power Method

TL;DR

This paper introduces a new power method-based algorithm for optimally clustering bipartite graphs in high-dimensional settings, extending previous work to more general cases and establishing minimax bounds.

Contribution

It develops a novel algorithm that handles general bipartite clustering with unequal community counts and derives optimal recovery conditions and bounds.

Findings

The algorithm achieves exact recovery under specified conditions.

It extends previous methods to the case where the number of communities differs between sides.

A minimax lower bound matches the upper bound up to a factor depending on K.

Abstract

Clustering bipartite graphs is a fundamental task in network analysis. In the high-dimensional regime where the number of rows $n_1$ and the number of columns $n_2$ of the associated adjacency matrix are of different order, existing methods derived from the ones used for symmetric graphs can come with sub-optimal guarantees. Due to increasing number of applications for bipartite graphs in the high dimensional regime, it is of fundamental importance to design optimal algorithms for this setting. The recent work of Ndaoud et al. (2022) improves the existing upper-bound for the misclustering rate in the special case where the columns (resp. rows) can be partitioned into $L = 2$ (resp. $K = 2$) communities. Unfortunately, their algorithm cannot be extended to the more general setting where $K \neq L \geq 2$. We overcome this limitation by introducing a new algorithm based on the power method. We derive conditions for exact recovery in the general setting where $K \neq L \geq 2$, and show that it recovers the result in Ndaoud et al. (2022). We also derive a minimax lower bound on the misclustering error when $K = L$ under a symmetric version of our model, which matches the corresponding upper bound up to a factor depending on $K$.