The game theoretic p-Laplacian and semi-supervised learning with few labels

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Abstract

We study the game theoretic p-Laplacian for semi-supervised learning on graphs, and show that it is well-posed in the limit of finite labeled data and infinite unlabeled data. In particular, we show that the continuum limit of graph-based semi-supervised learning with the game theoretic p-Laplacian is a weighted version of the continuous p-Laplace equation. We also prove that solutions to the graph p-Laplace equation are approximately Hölder continuous with high probability. Our proof uses the viscosity solution machinery and the maximum principle on a graph.

Original languageEnglish (US)
Pages (from-to)301-330
Number of pages30
JournalNonlinearity
Volume32
Issue number1
DOIs
StatePublished - Jan 2019

Bibliographical note

Funding Information:
The author gratefully acknowledges the support of NSF-DMS grant 1713691. The author is also grateful to the anonymous referees, whose suggestions have greatly improved the paper.

Publisher Copyright:
© 2018 IOP Publishing Ltd & London Mathematical Society.

Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

Keywords

  • consistency
  • continuum limit
  • game theoretic p-Laplacian
  • maximum principle
  • probability
  • semi-supervised learning
  • viscosity solutions

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