Calculus of Variations and Geometric Measure Theory

R. Cristoferi - M. Thorpe

Large Data Limit for a Phase Transition Model with the $p$-Laplacian on Point Clouds

created by cristoferi on 23 Feb 2018
modified on 14 Sep 2021


Published Paper

Inserted: 23 feb 2018
Last Updated: 14 sep 2021

Journal: European Journal of Applied Mathematics
Volume: 31
Pages: 185-231
Year: 2018
Links: Published Paper


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.