Finding Planted Subgraphs with Few Eigenvalues using the Schur-Horn Relaxation

TL;DR

This paper introduces a new convex relaxation method based on spectral majorization inequalities for detecting planted subgraphs with few eigenvalues, extending previous techniques for specific structured subgraphs.

Contribution

It presents a novel semidefinite relaxation approach that efficiently finds planted subgraphs with few eigenvalues, generalizing earlier convex methods for planted cliques.

Findings

Effective at identifying subgraphs with few eigenvalues

Generalizes convex relaxation techniques for planted cliques

Relies on spectrally comonotone matrices for analysis

Abstract

Extracting structured subgraphs inside large graphs - often known as the planted subgraph problem - is a fundamental question that arises in a range of application domains. This problem is NP-hard in general, and as a result, significant efforts have been directed towards the development of tractable procedures that succeed on specific families of problem instances. We propose a new computationally efficient convex relaxation for solving the planted subgraph problem; our approach is based on tractable semidefinite descriptions of majorization inequalities on the spectrum of a symmetric matrix. This procedure is effective at finding planted subgraphs that consist of few distinct eigenvalues, and it generalizes previous convex relaxation techniques for finding planted cliques. Our analysis relies prominently on the notion of spectrally comonotone matrices, which are pairs of symmetric matrices that can be transformed to diagonal matrices with sorted diagonal entries upon conjugation by the same orthogonal matrix.