[Paper Review] Only distances are required to reconstruct submanifolds

Jean‐Daniel Boissonnat, Ramsay Dyer+2 more|arXiv (Cornell University)|Oct 26, 2014

Topological and Geometric Data Analysis46 references4 citations

TL;DR

This paper presents a novel algorithm for reconstructing submanifolds in Euclidean space using only a distance matrix as input, without requiring explicit coordinates or complex data structures like Delaunay complexes. By introducing the concept of power protection and leveraging the witness complex with weighted points, the method achieves faithful topological and geometric reconstruction with complexity exponential in the intrinsic dimension but linear in the ambient dimension.

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.

Motivation & Objective

• To develop a manifold reconstruction algorithm that operates solely from a distance matrix, eliminating the need for explicit point coordinates.
• To avoid maintaining complex geometric data structures such as Voronoi diagrams or Delaunay complexes, which are computationally expensive and dimension-sensitive.
• To achieve faithful reconstruction—topologically correct and geometrically close—to the underlying submanifold using only elementary data structures.
• To reduce computational complexity dependence on ambient dimension while preserving exponential dependence only on the intrinsic manifold dimension.
• To establish theoretical guarantees for stability and correctness under minimal sampling assumptions using a novel notion of power protection.

Proposed method

• Introduces the concept of power protection for Delaunay simplices in weighted point sets, extending stability results from Delaunay triangulations to the witness complex framework.
• Employs the witness complex as the core data structure, where a simplex is included if a witness point sees it as 'visible' via distance comparisons, avoiding coordinate-dependent predicates.
• Uses a lifting map into the space of spheres (Laguerre geometry) to analyze power protection and stability, enabling robustness to perturbations.
• Applies a weighted witness complex construction where point weights are derived from local geometry, ensuring that only well-protected simplices are retained.
• Relies on the stability of power protection to ensure that small perturbations in the input do not disrupt the topological structure of the reconstructed complex.
• Uses geometric projections and angle bounds (e.g., between tangent spaces and normal flats) to prove that the reconstructed complex is homeomorphic to the original manifold.

Experimental results

Research questions

• RQ1 Can a faithful reconstruction of a submanifold be achieved using only pairwise distances, without explicit coordinates?
• RQ2 Is it possible to avoid constructing full Delaunay or Voronoi complexes while still ensuring topological correctness and geometric fidelity?
• RQ3 Can the stability of Delaunay triangulations be extended to the witness complex framework via a new notion of protection?
• RQ4 Does the algorithm’s complexity depend only exponentially on the intrinsic dimension and linearly on the ambient dimension?
• RQ5 Can power protection in the space of spheres ensure robustness to sampling noise and geometric perturbations?

Key findings

• The proposed algorithm reconstructs a submanifold faithfully—homeomorphic and geometrically close—using only the distance matrix as input.
• The algorithm avoids explicit coordinate computation and does not require maintaining Delaunay or Voronoi complexes, significantly reducing structural complexity.
• The computational complexity depends exponentially on the intrinsic dimension $ m $ of the manifold and linearly on the ambient dimension $ d $, which is optimal in this context.
• The method introduces power protection for weighted Delaunay simplices, enabling stability guarantees in the witness complex framework.
• The witness complex constructed via power protection is shown to be a faithful approximation of the manifold, with a well-defined geometric correspondence via projection maps.
• Theoretical analysis proves that under appropriate sampling and reach conditions, the reconstructed complex is a diffeomorphic copy of the original manifold within local neighborhoods.