Reconstruction of manifold embeddings into Euclidean spaces via intrinsic distances

¹ HSE University and Institute for Information Transmission Problems, Moscow, Russian Federation
² Weierstrass Institute of Applied Analysis and Stochastics and Humboldt University, Berlin, Germany and HSE University and Institute for Information Transmission Problems, Moscow, Russian Federation
³ St.Petersburg Branch of the Steklov Mathematical Institute of the Russian Academy of Sciences, St.Petersburg, Russian Federation and Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy and HSE University, Moscow, Russian Federation
⁴ Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

^* Corresponding author: stepanov.eugene@gmail.com

Received: 24 January 2022
Accepted: 3 December 2023

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).