Manifold learning in metric spaces

• URL: http://arxiv.org/abs/2503.16187v1
• Date: Thu, 20 Mar 2025 14:37:40 GMT
• Title: Manifold learning in metric spaces
• Authors: Liane Xu, Amit Singer,
• Abstract summary: Laplacian-based methods are popular for dimensionality reduction of data lying in $mathbbRN$.<n>We provide a framework that generalizes the problem of manifold learning to metric spaces and study when a metric satisfies sufficient conditions for the pointwise convergence of the graph Laplacian.
• Score: 4.849550522970841
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Laplacian-based methods are popular for dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance approximates the geodesic distance on the underlying submanifold which the data are assumed to lie on. However, for some applications, other metrics, such as the Wasserstein distance, may provide a more appropriate notion of distance than the Euclidean distance. We provide a framework that generalizes the problem of manifold learning to metric spaces and study when a metric satisfies sufficient conditions for the pointwise convergence of the graph Laplacian.

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