Only distances are required to reconstruct submanifolds

TL;DR

This paper introduces a novel algorithm for reconstructing submanifolds in Euclidean space using only distance data, avoiding complex data structures and explicit point coordinates, with complexity depending on the manifold's intrinsic dimension.

Contribution

The first algorithm to faithfully reconstruct submanifolds using solely distance matrices, leveraging the stability of power protection and the witness complex.

Findings

Reconstruction relies only on distance matrices, no explicit coordinates needed.

Algorithm complexity depends exponentially on intrinsic dimension, linearly on ambient space.

Introduces the concept of stability of power protection for manifold reconstruction.

Abstract

In this paper, we give the first algorithm that outputs a faithful reconstruction of a submanifold of Euclidean space without maintaining or even constructing complicated data structures such as Voronoi diagrams or Delaunay complexes. Our algorithm uses the witness complex and relies on the stability of power protection, a notion introduced in this paper. The complexity of the algorithm depends exponentially on the intrinsic dimension of the manifold, rather than the dimension of ambient space, and linearly on the dimension of the ambient space. Another interesting feature of this work is that no explicit coordinates of the points in the point sample is needed. The algorithm only needs the distance matrix as input, i.e., only distance between points in the point sample as input.