Provably Finding a Hidden Dense Submatrix among Many Planted Dense Submatrices via Convex Programming

TL;DR

This paper develops convex programming methods to reliably identify a hidden dense submatrix within a larger matrix containing multiple such submatrices, extending previous work to more realistic scenarios.

Contribution

It generalizes existing convex relaxation techniques to detect multiple dense submatrices in matrices with complex structures, providing polynomial-time recovery guarantees.

Findings

Sufficient conditions for exact recovery in polynomial time.

Phase transition phenomena observed in numerical experiments.

Validation on real-world networks confirms theoretical predictions.

Abstract

We consider the densest submatrix problem, which seeks the submatrix of fixed size of a given binary matrix that contains the most nonzero entries. This problem is a natural generalization of fundamental problems in combinatorial optimization, e.g., the densest subgraph, maximum clique, and maximum edge biclique problems, and has wide application the study of complex networks. Much recent research has focused on the development of sufficient conditions for exact solution of the densest submatrix problem via convex relaxation. The vast majority of these sufficient conditions establish identification of the densest submatrix within a graph containing exactly one large dense submatrix hidden by noise. The assumptions of these underlying models are not observed in real-world networks, where the data may correspond to a matrix containing many dense submatrices of varying sizes. We extend and generalize these results to the more realistic setting where the input matrix may contain \emph{many} large dense subgraphs. Specifically, we establish sufficient conditions under which we can expect to solve the densest submatrix problem in polynomial time for random input matrices sampled from a generalization of the stochastic block model. Moreover, we also provide sufficient conditions for perfect recovery under a deterministic adversarial. Numerical experiments involving randomly generated problem instances and real-world collaboration and communication networks are used empirically to verify the theoretical phase-transitions to perfect recovery given by these sufficient conditions.