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

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.