Publication 18-CNA-004
Large Data Limit for a Phase Transition Model with the $p$-Laplacian on Point Clouds
Riccardo Cristoferi
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh PA 15213-3890 USA
rcristof@andrew.cmu.edu
Matthew Thorpe
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
Cambridge, UK
m.thorpe@maths.cam.ac.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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