Low-rank optimization for semidefinite convex problems

TL;DR

This paper introduces a low-rank factorization approach for solving semidefinite convex problems efficiently by reformulating them on a quotient manifold and applying second-order optimization methods.

Contribution

It presents a novel algorithm leveraging matrix factorization and manifold geometry to solve semidefinite programs with low-rank solutions more effectively.

Findings

Effective for low-rank solutions

Demonstrated on graph cut and PCA problems

Provides conditions for equivalence with original problem

Abstract

We propose an algorithm for solving nonlinear convex programs defined in terms of a symmetric positive semidefinite matrix variable $X$. This algorithm rests on the factorization $X=Y Y^T$, where the number of columns of Y fixes the rank of $X$. It is thus very effective for solving programs that have a low rank solution. The factorization $X=Y Y^T$ evokes a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second order optimization method. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. The efficiency of the proposed algorithm is illustrated on two applications: the maximal cut of a graph and the sparse principal component analysis problem.