Calculus of Variations and Geometric Measure Theory
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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 24 Sep 2018

[BibTeX]

Accepted Paper

Inserted: 23 feb 2018
Last Updated: 24 sep 2018

Journal: European Journal of Applied Mathematics
Year: 2018

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.


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