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

Published online by Cambridge University Press:  14 November 2018

R. CRISTOFERI
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh, Scotland, EH14 4AS, UK e-mail: [email protected]
M. THORPE
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK e-mail: [email protected]

Abstract

The consistency of a non-local 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 functions, and the p-Laplacian, that enforces regularity. Under appropriate scaling between the two terms, minimisers exhibit a phase transition on the order of ɛ = ɛn, where n is the number of data points. We study the large data asymptotics, i.e. as n → ∝, in the regime where ɛn → 0. The mathematical tool used to address this question is Γ-convergence. It is proved that the discrete model converges to a weighted anisotropic perimeter.

Type
Papers
Copyright
© Cambridge University Press 2018 

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Footnotes

The research of R. C. was funded by National Science Foundation under Grant No. DMS-1411646. Part of the research of M. T. was funded by the National Science Foundation under Grant No. CCT-1421502.

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