A simple inverse power method for balanced graph cut

TL;DR

This paper introduces the simple inverse power (SIP) method for balanced graph cut that features an explicit analytic solution for its inner subproblem, leading to faster convergence and improved efficiency over existing methods.

Contribution

The paper develops a novel SIP method with an explicit solution for the inner subproblem, enabling local convergence and applicability to a new ternary balanced cut problem.

Findings

SIP is significantly faster than the inverse power method.

SIP achieves comparable solution quality to existing methods.

SIP-perturb outperforms Gurobi in cost and solution quality.

Abstract

The existing inverse power ($\mathbf{IP}$) method for solving the balanced graph cut lacks local convergence and its inner subproblem requires a nonsmooth convex solver. To address these issues, we develop a simple inverse power ($\mathbf{SIP}$) method using a novel equivalent continuous formulation of the balanced graph cut, and its inner subproblem allows an explicit analytic solution, which is the biggest advantage over $\mathbf{IP}$ and constitutes the main reason why we call it $\mathit{simple}$. By fully exploiting the closed-form of the inner subproblem solution, we design a boundary-detected subgradient selection with which $\mathbf{SIP}$ is proved to be locally converged. We show that $\mathbf{SIP}$ is also applicable to a new ternary valued $\theta$-balanced cut which reduces to the balanced cut when $\theta=1$. When $\mathbf{SIP}$ reaches its local optimum, we seamlessly transfer to solve the $\theta$-balanced cut within exactly the same iteration algorithm framework and thus obtain $\mathbf{SIP}$-$\mathbf{perturb}$ -- an efficient local breakout improvement of $\mathbf{SIP}$, which transforms some ``partitioned" vertices back to the ``un-partitioned" ones through the adjustable $\theta$. Numerical experiments on G-set for Cheeger cut and Sparsest cut demonstrate that $\mathbf{SIP}$ is significantly faster than $\mathbf{IP}$ while maintaining approximate solutions of comparable quality, and $\mathbf{SIP}$-$\mathbf{perturb}$ outperforms $\mathtt{Gurobi}$ in terms of both computational cost and solution quality.