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Reconstruction of manifold embeddings into Euclidean spaces via intrinsic distances

Nikita Puchkin, Vladimir Spokoiny, Eugene Stepanov, Dario Trevisan

TL;DR

This work provides 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).

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).

Reconstruction of manifold embeddings into Euclidean spaces via intrinsic distances

TL;DR

This work provides 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).

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).

Paper Structure

This paper contains 10 sections, 11 theorems, 97 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Main existing methods
  4. Multidimensional scaling (MDS)
  5. Maximum variance unfolding (MVU)
  6. Variational setting
  7. Discrete variational setting and algorithm
  8. How to compute Čech cohomologies
  9. Numerical experiments
  10. Auxiliary lemmata on sets of positive reach

Key Result

Theorem 4.1

Let $M\subset \mathbb{R}^n$ be compact and connected by rectifiable arcs, and there exist $\varepsilon_0>0$, $C_1>0$ such that for all $(x,y)\in M\times M$ satisfying $d_M(x,y)\leq \varepsilon_0$. Denote For an $x_0\in M$, $\Sigma_k\subset M$ a sequence of closed sets satisfying $\Sigma_k\to M$ as $k\to\infty$ in the sense of the Hausdorff distance, and a $C_2\in (0, \bar{C}_2]$ set Then the fo

Figures (4)

  • Figure 7.1: Reconstruction of a line segment from pairwise distances. The average distance error is $9 \cdot 10^{-4}$.
  • Figure 7.2: Reconstruction of a unit sphere from pairwise distances. Top line, columns 1 and 2: points on a grid on the sphere. Top line, columns 3 and 4: the recovered points of the unit sphere. Bottom line, columns 1 and 2: points drawn from the uniform distribution on unit sphere. Bottom line, columns 3 and 4: the recovered points from approximate geodesic distances.
  • Figure 7.3: Reconstruction of the Swiss Roll from pairwise distances. Top line: sample points in $\mathbb{R}^3$ (left), sample points, projection on the XOZ plane (center), reconstructed Swiss Roll (right). Bottom line: sample points in $\mathbb{R}^3$ (left), sample points, projection on the XOZ plane (center), embedding into $\mathbb{R}^2$ (right).
  • Figure 7.4: Reconstruction of Clifford torus from pairwise distances. Top line: projections of the original point cloud. Bottom line: projections of the points after embedding.

Theorems & Definitions (30)

  • Example 3.1
  • Example 3.2
  • Theorem 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • proof : Proof of Theorem \ref{['th_Vdistest1']}
  • Lemma 4.5
  • proof
  • Lemma 4.6
  • ...and 20 more