On the low-rank approach for semidefinite programs arising in synchronization and community detection

TL;DR

This paper investigates the effectiveness of a low-rank heuristic for solving large semidefinite programs in synchronization and community detection, providing theoretical insights into its empirical success.

Contribution

It offers theoretical guarantees explaining why the low-rank heuristic performs well in these specific applications.

Findings

The heuristic is theoretically justified in certain synchronization problems.

It explains the empirical success of low-rank approaches in community detection.

Provides bounds and conditions under which the heuristic is effective.

Abstract

To address difficult optimization problems, convex relaxations based on semidefinite programming are now common place in many fields. Although solvable in polynomial time, large semidefinite programs tend to be computationally challenging. Over a decade ago, exploiting the fact that in many applications of interest the desired solutions are low rank, Burer and Monteiro proposed a heuristic to solve such semidefinite programs by restricting the search space to low-rank matrices. The accompanying theory does not explain the extent of the empirical success. We focus on Synchronization and Community Detection problems and provide theoretical guarantees shedding light on the remarkable efficiency of this heuristic.