Compressive Sensing for cut improvement and local clustering

TL;DR

This paper introduces ClusterPursuit, a novel approach that formulates cut improvement as a sparse recovery problem, enabling fast algorithms with statistical guarantees for graph clustering tasks.

Contribution

It presents a new method, ClusterPursuit, leveraging compressive sensing algorithms for cut improvement, and applies it to local and semi-supervised clustering with proven guarantees.

Findings

Fast and theoretically sound cut improvement method

Effective in probabilistic graph models like Stochastic Block Model

Validated through extensive numerical experiments

Abstract

We show how one can phrase the cut improvement problem for graphs as a sparse recovery problem, whence one can use algorithms originally developed for use in compressive sensing (such as SubspacePursuit or CoSaMP) to solve it. We show that this approach to cut improvement is fast, both in theory and practice and moreover enjoys statistical guarantees of success when applied to graphs drawn from probabilistic models such as the Stochastic Block Model. Using this new cut improvement approach, which we call ClusterPursuit, as an algorithmic primitive we then propose new methods for local clustering and semi-supervised clustering, which enjoy similar guarantees of success and speed. Finally, we verify the promise of our approach with extensive numerical benchmarking.