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

Part of: Nonparametric inference Existence theories

Published online by Cambridge University Press:  14 November 2018

R. CRISTOFERI  and
M. THORPE
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]
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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.

Keywords

Ginzburg–Landau functionalPDEs on graphsnon-local variational problemsphase transitionsΓ-convergence

MSC classification

Primary: 49J55: Problems involving randomness
Secondary: 49J45: Methods involving semicontinuity and convergence; relaxation 62G20: Asymptotic properties
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 2 , April 2020 , pp. 185 - 231
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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