Correlation Clustering with Low-Rank Matrices

TL;DR

This paper investigates the computational complexity of correlation clustering with low-rank matrices, proving polynomial-time solvability for positive semidefinite cases and developing an efficient algorithm based on zonotope vertex enumeration.

Contribution

It establishes the polynomial-time solvability of correlation clustering for low-rank positive semidefinite matrices and introduces a novel algorithm leveraging zonotope vertex enumeration.

Findings

Polynomial-time solution for low-rank positive semidefinite matrices.

NP-hardness persists with negative eigenvalues.

Algorithm demonstrates effectiveness on synthetic and real data.

Abstract

Correlation clustering is a technique for aggregating data based on qualitative information about which pairs of objects are labeled 'similar' or 'dissimilar.' Because the optimization problem is NP-hard, much of the previous literature focuses on finding approximation algorithms. In this paper we explore how to solve the correlation clustering objective exactly when the data to be clustered can be represented by a low-rank matrix. We prove in particular that correlation clustering can be solved in polynomial time when the underlying matrix is positive semidefinite with small constant rank, but that the task remains NP-hard in the presence of even one negative eigenvalue. Based on our theoretical results, we develop an algorithm for efficiently "solving" low-rank positive semidefinite correlation clustering by employing a procedure for zonotope vertex enumeration. We demonstrate the effectiveness and speed of our algorithm by using it to solve several clustering problems on both synthetic and real-world data.