Large data limit of the MBO scheme for data clustering: convergence of the dynamics

TL;DR

This paper proves that the MBO scheme for data clustering converges to a mean curvature flow solution, providing theoretical insights into parameter choices based on sample size and interaction width.

Contribution

It introduces a new convergence result for the MBO scheme on random geometric graphs, linking it to mean curvature flow and guiding practical parameter selection.

Findings

Convergence of MBO scheme to viscosity solutions of mean curvature flow.

Quantitative estimates for heat operators on random geometric graphs.

Guidelines for selecting parameters based on sample size and interaction width.

Abstract

We prove that the dynamics of the MBO scheme for data clustering converge to a viscosity solution to mean curvature flow. The main ingredients are (i) a new abstract convergence result based on quantitative estimates for heat operators and (ii) the derivation of these estimates in the setting of random geometric graphs. To implement the scheme in practice, two important parameters are the number of eigenvalues for computing the heat operator and the step size of the scheme. The results of the current paper give a theoretical justification for the choice of these parameters in relation to sample size and interaction width.