Laplace Learning in Wasserstein Space
This work addresses the challenge of applying semi-supervised learning to high-dimensional data under the manifold hypothesis, representing an incremental extension of classical methods to a new mathematical framework.
The paper tackles the problem of extending graph-based semi-supervised learning to infinite-dimensional settings by investigating Laplace Learning in Wasserstein space, proving variational convergence and characterizing the Laplace-Beltrami operator, with numerical experiments showing consistent classification performance on benchmark datasets.
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.