Wasserstein Soft Label Propagation on Hypergraphs: Algorithm and Generalization Error Bounds

TL;DR

This paper introduces a novel semi-supervised learning algorithm for hypergraphs using Wasserstein barycenters to propagate soft labels, and provides theoretical generalization error bounds within a PAC framework.

Contribution

It reformulates label propagation on hypergraphs via Wasserstein barycenters and establishes generalization bounds based on algorithmic stability, extending Wasserstein propagation theory.

Findings

Algorithm achieves effective soft label propagation on hypergraphs.

Provides PAC-based generalization error bounds for the method.

Enhances understanding of Wasserstein propagation on complex graph structures.

Abstract

Inspired by recent interests of developing machine learning and data mining algorithms on hypergraphs, we investigate in this paper the semi-supervised learning algorithm of propagating "soft labels" (e.g. probability distributions, class membership scores) over hypergraphs, by means of optimal transportation. Borrowing insights from Wasserstein propagation on graphs [Solomon et al. 2014], we re-formulate the label propagation procedure as a message-passing algorithm, which renders itself naturally to a generalization applicable to hypergraphs through Wasserstein barycenters. Furthermore, in a PAC learning framework, we provide generalization error bounds for propagating one-dimensional distributions on graphs and hypergraphs using 2-Wasserstein distance, by establishing the \textit{algorithmic stability} of the proposed semi-supervised learning algorithm. These theoretical results also shed new lights upon deeper understandings of the Wasserstein propagation on graphs.