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Publication 18-CNA-004

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

R. Cristoferi
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh PA 15213-3890 USA

Matthew Thorpe
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
Cambridge, UK

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 $\varepsilon$ = $\varepsilon_n$ where $n$ is the number of data points. We study the large data asymptotics, i.e. as $n \to \infty$, in the regime where $\varepsilon_n \to 0$. 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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