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

Riccardo Cristoferi, Matthew Thorpe

Research output: Contribution to journalArticlepeer-review

Abstract

he 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 = ɛ_n, where n is the number of data points. We study the large data asymptotics, i.e. as n → ∝, in the regime where ɛ_n → 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.
Original languageEnglish
Pages (from-to)185-231
JournalEuropean Journal of Applied Mathematics
Volume31
Issue number2
Publication statusPublished - 14 Nov 2018

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