Laplace Learning in Wasserstein Space

TL;DR

This paper extends graph-based semi-supervised learning, specifically Laplace Learning, into the Wasserstein space, providing theoretical convergence results and validating the approach with experiments on high-dimensional data.

Contribution

It introduces a novel theoretical framework for Laplace Learning in Wasserstein space, including convergence analysis and operator characterization.

Findings

The discrete graph p-Dirichlet energy converges to its continuum limit.

The Laplace-Beltrami operator on Wasserstein submanifolds is characterized.

Experimental results show consistent classification performance in high dimensions.

Abstract

The manifold hypothesis posits that high-dimensional data typically resides on low-dimensional sub spaces. In this paper, we assume manifold hypothesis to investigate graph-based semi-supervised learning methods. In particular, we examine Laplace Learning in the Wasserstein space, extending the classical notion of graph-based semi-supervised learning algorithms from finite-dimensional Euclidean spaces to an infinite-dimensional setting. To achieve this, we prove variational convergence of a discrete graph p- Dirichlet energy to its continuum counterpart. In addition, we characterize the Laplace-Beltrami operator on asubmanifold of the Wasserstein space. Finally, we validate the proposed theoretical framework through numerical experiments conducted on benchmark datasets, demonstrating the consistency of our classification performance in high-dimensional settings.