Minimal Dirichlet energy partitions for graphs

TL;DR

This paper introduces a novel non-convex graph partitioning method based on Dirichlet eigenvalues, with a new algorithm that converges to local minima, applicable to various datasets including synthetic data and MNIST digits.

Contribution

It proposes a new Dirichlet energy-based partitioning objective, a relaxed formulation, and a rearrangement algorithm with proven convergence, extending to semi-supervised clustering.

Findings

Algorithm converges in finite steps

Effective on synthetic, MNIST, and manifold data

Provides natural cluster representatives

Abstract

Motivated by a geometric problem, we introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show is strictly decreasing and converges in a finite number of iterations to a local minimum of the relaxed objective function. Our method is applied to several clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations. The model has a semi-supervised extension and provides a natural representative for the clusters as well.