Manifold learning in metric spaces
This work addresses the need for more flexible distance metrics in manifold learning, which is incremental as it extends existing methods to new contexts.
The paper tackles the problem of generalizing manifold learning methods, like Laplacian-based dimensionality reduction, from Euclidean spaces to arbitrary metric spaces, such as those using the Wasserstein distance, and provides a framework with conditions for the pointwise convergence of the graph Laplacian.
Laplacian-based methods are popular for the dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance locally 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.