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Sensor network localization has a benign landscape after low-dimensional relaxation

Christopher Criscitiello, Andrew D. McRae, Quentin Rebjock, Nicolas Boumal

Abstract

We consider the sensor network localization problem, which is closely related to multidimensional scaling and Euclidean distance matrix completion. Given a ground truth configuration of $n$ points in $\mathbb{R}^\ell$, we observe a subset of the pairwise distances and aim to recover the underlying configuration (up to rigid transformations). We show with a simple counterexample that the associated optimization problem is nonconvex and may admit spurious local minimizers, even when all distances are known. Yet, inspired by numerical experiments, we argue that all second-order critical points become global minimizers when the problem is relaxed by optimizing over configurations in dimension $k > \ell$. Specifically, we show this for two settings, both when all pairwise distances are known: (1) for arbitrary ground truth points, and $k= O(\sqrt{\ell n})$, and: (2) for isotropic random ground truth points, and $k = O(\ell + \log n)$. To prove these results, we identify and exploit key properties of the linear map which sends inner products to squared distances.

Sensor network localization has a benign landscape after low-dimensional relaxation

Abstract

We consider the sensor network localization problem, which is closely related to multidimensional scaling and Euclidean distance matrix completion. Given a ground truth configuration of points in , we observe a subset of the pairwise distances and aim to recover the underlying configuration (up to rigid transformations). We show with a simple counterexample that the associated optimization problem is nonconvex and may admit spurious local minimizers, even when all distances are known. Yet, inspired by numerical experiments, we argue that all second-order critical points become global minimizers when the problem is relaxed by optimizing over configurations in dimension . Specifically, we show this for two settings, both when all pairwise distances are known: (1) for arbitrary ground truth points, and , and: (2) for isotropic random ground truth points, and . To prove these results, we identify and exploit key properties of the linear map which sends inner products to squared distances.

Paper Structure

This paper contains 37 sections, 18 theorems, 95 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Mathematical background for SNL
  4. Notation
  5. The optimization problem
  6. Properties of the EDM map
  7. Eigenanalysis of $\Delta^* \circ \Delta$.
  8. The restricted isometry property fails.
  9. The EDM map as a structured perturbation of the identity
  10. Derivatives and optimality conditions
  11. Useful consequence of 2-criticality.
  12. $2$-critical rank-deficient configurations are optimal
  13. Landscape for all ground truths, $k \approx \sqrt{n}$
  14. Exploiting first-order criticality
  15. Second-order criticality and descent directions
  16. ...and 22 more sections

Key Result

Theorem 1.1

If $E$ in eq:snl is complete and $k \gtrsim \ell + \sqrt{\ell n}$, then eq:snl has benign landscape.

Figures (7)

  • Figure 1: Relaxing the dimension (that is, optimizing in dimension $k$ larger than the ground truth dimension $\ell$) empirically increases the probability that an optimization algorithm (here, trust regions) recovers the ground truth for \ref{['eq:snl']} from a random initialization. This is apparent for all Erdős--Rényi densities, and especially for sparse graphs. The black curve marks the fraction of measurement graphs that are connected. Measurements here are noiseless; however, these empirical results are robust to moderate levels of noise (see Appendix \ref{['app:noise']}). Details are in Section \ref{['sec:numerics']}.
  • Figure 2: Ground truth (left) and spurious configuration (right) for $n = 7$, $\ell = 2$, and $k = 2$. The latter has two overlapping points at $(0, 1)$. Observe that when we relax to $k = 3$, it becomes possible to transform the spurious configuration (right) into the ground truth (left) while decreasing the cost \ref{['eq:snl']}: rotate one of the $(0,1)$ points around the horizontal axis---out of the plane---while keeping the other six points fixed.
  • Figure 3: The points on the horizontal axis are the eigenvalues of $P$. The corresponding points on the curve $- a t + b t^2 + n t^3$ are the eigenvalues of $- a P + b P^2 + n P^3$. See the proof of Theorem \ref{['th:sqrt-n']} in Section \ref{['sectioneigenvalueinterlacing']}.
  • Figure 4: Success rates for ground truth recovery. The blue curves never attain a success rate of 1.
  • Figure 5: Success rates for global optimality. The blue curves never attain a success rate of 1, except when the density is zero.
  • ...and 2 more figures

Theorems & Definitions (35)

  • Theorem 1.1: All ground truths
  • Theorem 1.2: Random ground truths
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof : Proof of Lemma \ref{['lem:cute']}
  • proof : Proof of Lemma \ref{['lem:cute']} based on randomized directions
  • Lemma 3.4
  • ...and 25 more