Laplace Learning in Wasserstein Space

2025-11-17

AI Summary

This paper addresses the failure of the Euclidean space assumption in semi-supervised learning for high-dimensional data. Methodologically, it extends graph Laplacian learning from finite-dimensional Euclidean spaces to the infinite-dimensional Wasserstein space—its first such formulation. Leveraging the manifold assumption, it rigorously characterizes the Laplace–Beltrami operator on submanifolds of the Wasserstein space and establishes a theoretically consistent classification framework via variational convergence analysis of discrete graph p-Dirichlet energies. Its key contribution lies in the novel integration of Wasserstein geometry, p-Dirichlet energy modeling, and variational convergence theory—overcoming traditional graph-based learning’s reliance on linear or finite-dimensional spaces. Experiments on multiple benchmark datasets demonstrate that the method achieves both theoretical consistency and robust classification performance in high-dimensional settings, offering a new paradigm for semi-supervised learning in non-Euclidean, infinite-dimensional spaces.

Technology Category

• Machine Learning: Semi-Supervised Learning
• Computer Vision: Learning & Optimization for CV
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Extending graph-based semi-supervised learning to infinite-dimensional Wasserstein space
• Proving variational convergence of discrete graph energy to continuum
• Characterizing Laplace-Beltrami operator on Wasserstein space submanifolds

Innovation

• Extends Laplace Learning to infinite-dimensional Wasserstein space
• Proves variational convergence of discrete graph energy
• Characterizes Laplace-Beltrami operator on Wasserstein submanifold