Large data limit for a phase transition model with the p-Laplacian on point clouds

TL;DR

This paper analyzes the large data limit of a nonlocal Ginzburg-Landau functional with p-Laplacian on point clouds, showing convergence to an anisotropic perimeter as data size grows.

Contribution

It establishes the large data asymptotics of a nonlocal anisotropic Ginzburg-Landau model with p-Laplacian, demonstrating convergence to a weighted anisotropic perimeter.

Findings

Discrete model converges to weighted anisotropic perimeter

Phase transition occurs at a specific scale of epsilon

Gamma-convergence used to analyze asymptotics

Abstract

The consistency of a nonlocal anisotropic Ginzburg-Landau type functional for data classification and clustering is studied. The Ginzburg-Landau objective functional combines a double well potential, that favours indicator valued function, and the $p$-Laplacian, that enforces regularity. Under appropriate scaling between the two terms minimisers exhibit a phase transition on the order of $\epsilon=\epsilon_n$ where $n$ is the number of data points. We study the large data asymptotics, i.e. as $n\to \infty$, in the regime where $\epsilon_n\to 0$. The mathematical tool used to address this question is $\Gamma$-convergence. In particular, it is proved that the discrete model converges to a weighted anisotropic perimeter.