Abstract
We consider the problem of reconstructing an embedding of a compact connected Riemannian manifold in a Euclidean space up to an almost isometry, given the information on intrinsic distances between points from its “sufficiently large” subset. This is one of the classical manifold learning problems. It happens that the most popular methods to deal with such a problem, with a long history in data science, namely, the classical Multidimensional scaling (MDS) and the Maximum variance unfolding (MVU), actually miss the point and may provide results very far from an isometry; moreover, they may even give no bi-Lipshitz embedding. We will provide an easy variational formulation of this problem, which leads to an algorithm always providing an almost isometric embedding with the distortion of original distances as small as desired (the parameter regulating the upper bound for the desired distortion is an input parameter of this algorithm).
Subject
Computational Mathematics,Control and Optimization,Control and Systems Engineering
Reference27 articles.
1. Manifold estimation and singular deconvolution under Hausdorff loss
2. Manifold Reconstruction Using Tangential Delaunay Complexes
3. Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold Reconstruction
4. Fefferman C., Ivanov S., Kurylev Y., Lassas M. and Narayanan H., Fitting a putative manifold to noisy data, in Proceedings of the 31st Conference on Learning Theory. Vol. 75 of Proceedings of Machine Learning Research. (2018) 688–720.