Dual-Bounded Nonlinear Optimal Transport for Size Constrained Min Cut Clustering

TL;DR

This paper introduces a novel dual-bounded nonlinear optimal transport formulation for the size constrained min cut clustering problem, along with a Frank-Wolfe based solution method that achieves fast convergence and improved stability.

Contribution

It reformulates the min cut problem as a dual-bounded nonlinear optimal transport problem and develops the DNF method, a Frank-Wolfe based algorithm with proven convergence for this class of problems.

Findings

DNF achieves state-of-the-art clustering accuracy.

DNF converges at a rate of (1/t) for convex problems.

The method requires no parameters and offers better stability.

Abstract

Min cut is an important graph partitioning method. However, current solutions to the min cut problem suffer from slow speeds, difficulty in solving, and often converge to simple solutions. To address these issues, we relax the min cut problem into a dual-bounded constraint and, for the first time, treat the min cut problem as a dual-bounded nonlinear optimal transport problem. Additionally, we develop a method for solving dual-bounded nonlinear optimal transport based on the Frank-Wolfe method (abbreviated as DNF). Notably, DNF not only solves the size constrained min cut problem but is also applicable to all dual-bounded nonlinear optimal transport problems. We prove that for convex problems satisfying Lipschitz smoothness, the DNF method can achieve a convergence rate of \(\mathcal{O}(\frac{1}{t})\). We apply the DNF method to the min cut problem and find that it achieves state-of-the-art performance in terms of both the loss function and clustering accuracy at the fastest speed, with a convergence rate of \(\mathcal{O}(\frac{1}{\sqrt{t}})\). Moreover, the DNF method for the size constrained min cut problem requires no parameters and exhibits better stability.