Noise-Stable Rigid Graphs for Euclidean Embedding
This work addresses the problem of robust Euclidean embedding for applications like wireless sensor network localization and satellite positioning, offering incremental improvements with new theoretical guarantees and algorithmic efficiency.
The paper tackles the problem of evaluating and improving Euclidean embedding algorithms for global structure reconstruction by introducing a new noise-stability criterion and proving its properties for classical MDS, providing theoretical guarantees for precision and bounds. It also proposes an algorithm to construct minimum-cost noise-stable spanning graphs, enabling reliable localization with linear edges and sublinear costs, and reducing reconstruction time complexity from O(n^3) to O(n).
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.