An efficient volume-preserving MBO scheme for data clustering and classification

TL;DR

This paper introduces a novel, efficient volume-preserving MBO scheme for data clustering and classification, leveraging constraints, order statistics, and variational analysis to improve computational complexity and accuracy.

Contribution

It develops a new constrained MBO algorithm with proven efficiency and complexity bounds, connecting it to volume-preserving mean curvature flow.

Findings

Algorithm is more efficient than existing methods.

Provides theoretical complexity bounds.

Connects scheme to volume-preserving mean curvature flow.

Abstract

We propose and study a novel efficient algorithm for clustering and classification tasks based on the famous MBO scheme. On the one hand, inspired by Jacobs et al. [J. Comp. Phys. 2018], we introduce constraints on the size of clusters leading to a linear integer problem. We prove that the solution to this problem is induced by a novel order statistic. This viewpoint allows us to develop exact and highly efficient algorithms to solve such constrained integer problems. On the other hand, we prove an estimate of the computational complexity of our scheme, which is better than any available provable bounds for the state of the art. This rigorous analysis is based on a variational viewpoint that connects this scheme to volume-preserving mean curvature flow in the big data and small time-step limit.