The $1$-Laplacian Cheeger Cut: Theory and Algorithms

TL;DR

This paper reviews the theory and algorithms for the graph 1-Laplacian Cheeger cut, proposing a cell descend framework that unifies and improves existing methods, with empirical comparisons on typical graphs.

Contribution

It introduces a novel cell descend framework for the 1-Laplacian Cheeger cut, unifying inverse power and steepest descent methods within a new theoretical approach.

Findings

The cell descend framework guarantees objective decrease.

The proposed methods outperform existing algorithms on benchmark graphs.

Empirical results demonstrate the effectiveness of the new algorithms.

Abstract

This paper presents a detailed review of both theory and algorithms for the Cheeger cut based on the graph $1$-Laplacian. In virtue of the cell structure of the feasible set, we propose a cell descend (CD) framework for achieving the Cheeger cut. While plugging the relaxation to guarantee the decrease of the objective value in the feasible set, from which both the inverse power (IP) method and the steepest descent (SD) method can also be recovered, we are able to get two specified CD methods. Comparisons of all these methods are conducted on several typical graphs.