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Structured Sampling for Robust Euclidean Distance Geometry

Chandra Kundu, Abiy Tasissa, HanQin Cai

TL;DR

This work tackles robust Euclidean distance geometry in a setting with anchor and target nodes where anchor–anchor distances are exact but anchor–target distances suffer sparse outliers. It introduces Structured Robust EDG, a localized pipeline that first denoises the corrupted anchor–target block $F$ with Robust PCA and then uses Nyström completion to recover the full Gram matrix $X$ from the available blocks, avoiding the missing $G$ block. The recovered Gram matrix is projected onto a rank-$r$ positive semidefinite space to yield coordinates up to rigid motion, enabling accurate point reconstruction with relatively few anchors. Experiments on synthetic sensor localization data and real protein structures demonstrate robust recovery under high corruption and scalability to large molecules, highlighting practical impact in localization and molecular structure analysis.

Abstract

This paper addresses the problem of estimating the positions of points from distance measurements corrupted by sparse outliers. Specifically, we consider a setting with two types of nodes: anchor nodes, for which exact distances to each other are known, and target nodes, for which complete but corrupted distance measurements to the anchors are available. To tackle this problem, we propose a novel algorithm powered by Nyström method and robust principal component analysis. Our method is computationally efficient as it processes only a localized subset of the distance matrix and does not require distance measurements between target nodes. Empirical evaluations on synthetic datasets, designed to mimic sensor localization, and on molecular experiments, demonstrate that our algorithm achieves accurate recovery with a modest number of anchors, even in the presence of high levels of sparse outliers.

Structured Sampling for Robust Euclidean Distance Geometry

TL;DR

This work tackles robust Euclidean distance geometry in a setting with anchor and target nodes where anchor–anchor distances are exact but anchor–target distances suffer sparse outliers. It introduces Structured Robust EDG, a localized pipeline that first denoises the corrupted anchor–target block with Robust PCA and then uses Nyström completion to recover the full Gram matrix from the available blocks, avoiding the missing block. The recovered Gram matrix is projected onto a rank- positive semidefinite space to yield coordinates up to rigid motion, enabling accurate point reconstruction with relatively few anchors. Experiments on synthetic sensor localization data and real protein structures demonstrate robust recovery under high corruption and scalability to large molecules, highlighting practical impact in localization and molecular structure analysis.

Abstract

This paper addresses the problem of estimating the positions of points from distance measurements corrupted by sparse outliers. Specifically, we consider a setting with two types of nodes: anchor nodes, for which exact distances to each other are known, and target nodes, for which complete but corrupted distance measurements to the anchors are available. To tackle this problem, we propose a novel algorithm powered by Nyström method and robust principal component analysis. Our method is computationally efficient as it processes only a localized subset of the distance matrix and does not require distance measurements between target nodes. Empirical evaluations on synthetic datasets, designed to mimic sensor localization, and on molecular experiments, demonstrate that our algorithm achieves accurate recovery with a modest number of anchors, even in the presence of high levels of sparse outliers.

Paper Structure

This paper contains 13 sections, 6 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Notation
  4. Related work
  5. Euclidean distance geometry and Nyström method
  6. Robust PCA
  7. Background
  8. Nyström approximation for distance matrices
  9. Proposed approach
  10. Numerical experiments
  11. Synthetic data experiments
  12. Protein structure recovery
  13. Conclusion

Figures (3)

  • Figure 1: Visual results for synthetic experiments: The figure illustrates the comparison between original points and estimated points. Here, we set $n=80$, $m=20$, $r=2$ and $\alpha = 0.2$.
  • Figure 2: Target structure 1PTQ (blue) compared to the numerically estimated structure (orange). The visualization corresponds to a single experimental realization with $m = 30$ anchor points and corruption fraction $\alpha = 0.2$ ($\mathrm{RMSE}$ = 0.65).
  • Figure 3: Target structure 1W2E (blue) compared to the numerically estimated structure (orange). This visualization represents a single experimental realization with $m = 30$ anchor points and a corruption fraction of $\alpha = 0.2$ ($\mathrm{RMSE}$ = $0.71$).