How to Round Subspaces: A New Spectral Clustering Algorithm

TL;DR

This paper introduces a novel spectral clustering algorithm capable of recovering near-integral solutions with subspace closeness guarantees, improving upon previous methods especially for unbalanced clusters.

Contribution

The paper presents a new spectral clustering algorithm that achieves closer subspace recovery without restrictions on cluster sizes, surpassing previous bounds.

Findings

Recovers $k$-partitions with subspace close to the original in spectral norm

Can find disjoint unions of bounded degree expanders approximating graphs

Improves approximation of sparsest $k$-partition under certain spectral conditions

Abstract

A basic problem in spectral clustering is the following. If a solution obtained from the spectral relaxation is close to an integral solution, is it possible to find this integral solution even though they might be in completely different basis? In this paper, we propose a new spectral clustering algorithm. It can recover a $k$-partition such that the subspace corresponding to the span of its indicator vectors is $O(\sqrt{opt})$ close to the original subspace in spectral norm with $opt$ being the minimum possible ($opt \le 1$ always). Moreover our algorithm does not impose any restriction on the cluster sizes. Previously, no algorithm was known which could find a $k$-partition closer than $o(k \cdot opt)$. We present two applications for our algorithm. First one finds a disjoint union of bounded degree expanders which approximate a given graph in spectral norm. The second one is for approximating the sparsest $k$-partition in a graph where each cluster have expansion at most $\phi_k$ provided $\phi_k \le O(\lambda_{k+1})$ where $\lambda_{k+1}$ is the $(k+1)^{st}$ eigenvalue of Laplacian matrix. This significantly improves upon the previous algorithms, which required $\phi_k \le O(\lambda_{k+1}/k)$.