### 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 language | English (US) |
---|---|

Pages (from-to) | 301-330 |

Number of pages | 30 |

Journal | Nonlinearity |

Volume | 32 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

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

### Cite this

**The game theoretic p-Laplacian and semi-supervised learning with few labels.** / Calder, Jeffrey W.

Research output: Contribution to journal › Article

*Nonlinearity*, vol. 32, no. 1, pp. 301-330. https://doi.org/10.1088/1361-6544/aae949

}

TY - JOUR

T1 - The game theoretic p-Laplacian and semi-supervised learning with few labels

AU - Calder, Jeffrey W

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - consistency

KW - continuum limit

KW - game theoretic p-Laplacian

KW - maximum principle

KW - probability

KW - semi-supervised learning

KW - viscosity solutions

UR - http://www.scopus.com/inward/record.url?scp=85059930987&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85059930987&partnerID=8YFLogxK

U2 - 10.1088/1361-6544/aae949

DO - 10.1088/1361-6544/aae949

M3 - Article

AN - SCOPUS:85059930987

VL - 32

SP - 301

EP - 330

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 1

ER -