Noise-Stable Rigid Graphs for Euclidean Embedding

TL;DR

This paper introduces a noise-stability criterion for rigidity theory, proves its applicability to cMDS, and develops algorithms for constructing noise-stable graphs and efficient point cloud reconstruction, enhancing Euclidean embedding reliability.

Contribution

It proposes a new noise-stability criterion for rigidity, proves cMDS's noise-stability, and introduces algorithms for minimal-cost noise-stable graphs and fast point cloud reconstruction.

Findings

cMDS is noise-stable under generic conditions

Constructed minimal-cost noise-stable spanning graphs

Reconstructed point clouds in linear time

Abstract

We proposed a new criterion \textit{noise-stability}, which revised the classical rigidity theory, for evaluation of MDS algorithms which can truthfully represent the fidelity of global structure reconstruction; then we proved the noise-stability of the cMDS algorithm in generic conditions, which provides a rigorous theoretical guarantee for the precision and theoretical bounds for Euclidean embedding and its application in fields including wireless sensor network localization and satellite positioning. Furthermore, we looked into previous work about minimum-cost globally rigid spanning subgraph, and proposed an algorithm to construct a minimum-cost noise-stable spanning graph in the Euclidean space, which enabled reliable localization on sparse graphs of noisy distance constraints with linear numbers of edges and sublinear costs in total edge lengths. Additionally, this algorithm also suggests a scheme to reconstruct point clouds from pairwise distances at a minimum of $O(n)$ time complexity, down from $O(n^3)$ for cMDS.