Clustering with Semidefinite Programming and Fixed Point Iteration

TL;DR

This paper presents a new clustering method based on semidefinite programming relaxation of Max k-Cut, utilizing fixed point iteration for improved rounding, leading to better clustering results.

Contribution

Introduces a novel clustering approach using SDP relaxation and fixed point iteration for rounding, enhancing clustering quality over traditional randomized methods.

Findings

Fixed point iteration improves clustering accuracy.

SDP relaxation effectively partitions data into k clusters.

Method outperforms randomized rounding in experiments.

Abstract

We introduce a novel method for clustering using a semidefinite programming (SDP) relaxation of the Max k-Cut problem. The approach is based on a new methodology for rounding the solution of an SDP relaxation using iterated linear optimization. We show the vertices of the Max k-Cut relaxation correspond to partitions of the data into at most k sets. We also show the vertices are attractive fixed points of iterated linear optimization. Each step of this iterative process solves a relaxation of the closest vertex problem and leads to a new clustering problem where the underlying clusters are more clearly defined. Our experiments show that using fixed point iteration for rounding the Max k-Cut SDP relaxation leads to significantly better results when compared to randomized rounding.