Strong consistency guarantees for clustering high-dimensional bipartite graphs with the spectral method

Abstract

We investigate the problem of clustering bipartite graphs using a simple spectral method within the framework of the Bipartite Stochastic Block Model (BiSBM), a popular model for bipartite graphs having a community structure. Our focus lies in the high-dimensional setting where the number $n_1$ of rows, and $n_2$ of columns, of the associated adjacency matrix differ significantly. A recent study by [4] has established a sufficient and necessary condition related to the sparsity level $p_{\text{max}}$ of the bipartite graph, enabling the recovery of the latent partition of the rows. In their work, [4] introduces an iterative method that extends the approach proposed by [26] to achieve the stated recovery goal. However, empirical results suggest that the subsequent refinement algorithm does not significantly enhance the performance of the spectral method, indicating that the spectral method achieves exact recovery within the same regime as the refinement method. We establish this claim by deriving new entrywise bounds on the eigenvectors of the similarity matrix utilized by the spectral method. Our analysis extends the framework of [23], which is limited to symmetric matrices with restricted dependencies. As a critical technical step, we also derive an improved concentration inequality tailored for similarity matrices.