Rank-sparsity decomposition for planted quasi clique recovery

TL;DR

This paper introduces a convex relaxation method based on rank-sparsity decomposition to recover planted quasi-cliques in graphs, providing theoretical guarantees and empirical validation for its effectiveness.

Contribution

It develops a new convex formulation for planted quasi-clique recovery and establishes theoretical conditions for exact recovery, extending previous clique detection methods.

Findings

The method successfully recovers planted quasi-cliques under certain conditions.

Theoretical bounds on dual matrix norms certify recovery guarantees.

Numerical experiments support the theoretical results.

Abstract

In this paper, we apply the Rank-Sparsity Matrix Decomposition to the planted Maximum Quasi-Clique Problem (MQCP). This problem has the planted Maximum Clique Problem (MCP) as a special case. The maximum clique problem is NP-hard. A Quasi-clique or $\gamma$-clique is a dense graph with the edge density of at least $\gamma$, where $\gamma \in (0, 1]$. The maximum quasi-clique problem seeks to find such a subgraph with the largest cardinality in a given graph. Our method of choice is the low-rank plus sparse matrix splitting technique. We present a theoretical basis for when our convex relaxation problem recovers the planted maximum quasi-clique. We derived a new bound on the norm of the dual matrix that certifies the recovery using $l_{\infty,2} norm. We showed that when certain conditions are met, our convex formulation recovers the planted quasi-clique exactly. The numerical experiments we performed corroborated our theory.