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Sample-Efficient Geometry Reconstruction from Euclidean Distances using Non-Convex Optimization

Ipsita Ghosh, Abiy Tasissa, Christian Kümmerle

TL;DR

A local convergence guarantee for a variant of iteratively reweighted least squares (IRLS) applies if a minimal random set of observed distances is provided and a restricted isometry property is established restricted to a tangent space of the manifold of symmetric rank-$r$ matrices given random Euclidean distance measurements.

Abstract

The problem of finding suitable point embedding or geometric configurations given only Euclidean distance information of point pairs arises both as a core task and as a sub-problem in a variety of machine learning applications. In this paper, we aim to solve this problem given a minimal number of distance samples. To this end, we leverage continuous and non-convex rank minimization formulations of the problem and establish a local convergence guarantee for a variant of iteratively reweighted least squares (IRLS), which applies if a minimal random set of observed distances is provided. As a technical tool, we establish a restricted isometry property (RIP) restricted to a tangent space of the manifold of symmetric rank-$r$ matrices given random Euclidean distance measurements, which might be of independent interest for the analysis of other non-convex approaches. Furthermore, we assess data efficiency, scalability and generalizability of different reconstruction algorithms through numerical experiments with simulated data as well as real-world data, demonstrating the proposed algorithm's ability to identify the underlying geometry from fewer distance samples compared to the state-of-the-art.

Sample-Efficient Geometry Reconstruction from Euclidean Distances using Non-Convex Optimization

TL;DR

A local convergence guarantee for a variant of iteratively reweighted least squares (IRLS) applies if a minimal random set of observed distances is provided and a restricted isometry property is established restricted to a tangent space of the manifold of symmetric rank- matrices given random Euclidean distance measurements.

Abstract

The problem of finding suitable point embedding or geometric configurations given only Euclidean distance information of point pairs arises both as a core task and as a sub-problem in a variety of machine learning applications. In this paper, we aim to solve this problem given a minimal number of distance samples. To this end, we leverage continuous and non-convex rank minimization formulations of the problem and establish a local convergence guarantee for a variant of iteratively reweighted least squares (IRLS), which applies if a minimal random set of observed distances is provided. As a technical tool, we establish a restricted isometry property (RIP) restricted to a tangent space of the manifold of symmetric rank- matrices given random Euclidean distance measurements, which might be of independent interest for the analysis of other non-convex approaches. Furthermore, we assess data efficiency, scalability and generalizability of different reconstruction algorithms through numerical experiments with simulated data as well as real-world data, demonstrating the proposed algorithm's ability to identify the underlying geometry from fewer distance samples compared to the state-of-the-art.

Paper Structure

This paper contains 27 sections, 12 theorems, 121 equations, 7 figures, 2 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. MatrixIRLS for Euclidean Distance Geometry
  5. Computational Considerations
  6. Theoretical Analysis
  7. Local Convergence Analysis of Algorithm \ref{['algo:MatrixIRLS']}
  8. Dual Basis Construction and Local Restricted Isometry Property on Tangent Spaces
  9. Numerical Experiments
  10. Synthetic Data
  11. Ill-conditioned Data
  12. Real Data
  13. Conclusion
  14. Limitations
  15. Dual Basis Construction
  16. ...and 12 more sections

Key Result

Theorem 4.3

Let $\mathbf{X}^0 \in S_n$ be a matrix of rank $r$ that is $\nu$-incoherent, and let $\mathcal{A}: S_n \rightarrow \mathop{\mathrm{\mathbb{R}}}\nolimits^{m+n}$ be the measurement operator corresponding to an index set $\Omega \subset \mathbb{I}$ of size $m=|\Omega|$ that is drawn uniformly without

Figures (7)

  • Figure 1: Success probabilities for recovery for different algorithms, given Gaussian ground truths $\mathbf{X}^0$ of different ranks, computed across $24$ instances.
  • Figure 2: Empirical success probabilities for recovery for different algorithms, given ill-conditioned ground truths $\mathbf{X}^0$ of different ranks, computed across $8$ instances.
  • Figure 3: Top: Relative error plot. Bottom: Runtime across iterates for one instance.
  • Figure 4: Performance of EDG completion algorithms over 24 instances on ill-conditioned data.
  • Figure 5: Reconstruction of Protein 1BPM molecule by MatrixIRLS with $0.5\%$ samples
  • ...and 2 more figures

Theorems & Definitions (24)

  • Definition 3.1: KuemmerleMayrinkVerdun-ICML2021
  • Definition 3.2
  • Definition 4.1: Coherence for Gram matrices in the EDG problem, tasissa18
  • Remark 4.2
  • Theorem 4.3: Local convergence of MatrixIRLS for EDG with Quadratic Rate
  • Lemma 4.4: Dual Basis Construction
  • Theorem 4.5: Restricted Isometry Property for Sampling Operator $\mathop{\mathrm{\mathcal{Q}_{\Omega}}}\nolimits$
  • proof : Proof of \ref{['lemma:dualbasis']}
  • Lemma B.1
  • proof
  • ...and 14 more