Calculus of Variations and Geometric Measure Theory

N. Puchkin - V. Spokoiny - E. Stepanov - D. Trevisan

Reconstruction of manifold embeddings into Euclidean spaces via intrinsic distances

created by stepanov on 24 Dec 2020
modified by trevisan on 05 Jul 2024


Accepted Paper

Inserted: 24 dec 2020
Last Updated: 5 jul 2024

Journal: ESAIM: COCV
Year: 2020

ArXiv: 2012.13770 PDF


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).

Keywords: manifold learning, multidimensional scaling, maximum variance unfolding, manifold embedding