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



Inserted: 24 dec 2020
Last Updated: 24 dec 2020

Year: 2020


We consider one of the classical manifold learning problems, that of reconstructing up to an almost isometry an embedding of a compact connected Riemannian manifold in a Euclidean space given the information on intrinsic distances between points from its almost dense subset. It will be shown that the most popular methods in data science to deal with such a problem, the classical Multidimensional scaling (MDS) and the Maximum variance unfolding (MVU) actually miss the point and may provide results very far from an isometry (and even may give no biLipshitz embedding). We will then provide an easy variational formulation of this problem which leads to an algorithm always providing an almost isometric imbedding with given controlled small distortion of original distances.

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