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

Riccardo Cristoferi, M. Thorpe

Research output: Contribution to journalArticle

1 Citation (Scopus)
15 Downloads (Pure)

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 = n, where n is the number of data points. We study the large data asymptotics, i.e. as n, in the regime where n 0. The mathematical tool used to address this question is "-convergence. It is proved that the discrete model converges to a weighted anisotropic perimeter.

Original languageEnglish
Pages (from-to)1-47
Number of pages47
JournalEuropean Journal of Applied Mathematics
Early online date14 Nov 2018
DOIs
Publication statusE-pub ahead of print - 14 Nov 2018

Keywords

  • "-convergence
  • Ginzburg-Landau functional
  • non-local variational problems
  • PDEs on graphs
  • phase transitions

ASJC Scopus subject areas

  • Applied Mathematics

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