Reconstruction of manifold embeddings into Euclidean spaces via intrinsic distances

TL;DR

This paper addresses the classical problem of reconstructing manifold embeddings in Euclidean space from intrinsic distances, proposing a variational approach that guarantees almost isometric embeddings with controllable distortion.

Contribution

It introduces a new variational formulation for manifold reconstruction that ensures near-isometric embeddings, overcoming limitations of traditional methods like MDS and MVU.

Findings

The proposed algorithm achieves embeddings with arbitrarily small distortion.

Traditional methods may fail to produce bi-Lipschitz embeddings.

The variational approach guarantees near-isometry in manifold reconstruction.

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).