Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local Geometry Determines Global Landscape in Low-rank Factorization for Synchronization

Shuyang Ling

TL;DR

This work addresses orthogonal group synchronization from noisy pairwise measurements and analyzes when the nonconvex BM factorization landscape is free of spurious local minima. It proves a deterministic, algebraic condition linking the required rank p to the condition number of the SDR certificate (Laplacian) at the global minimizer, ensuring benignity for d = 1 and d ≥ 2 under near-SDR regimes. The results extend to robustness against monotone adversaries and apply across Z2 synchronization, SBMs, O(d) synchronization, and generalized Procrustes, delivering near-optimal noise thresholds and broad practical implications for scalable SDP solvers. Overall, the paper provides a unifying, geometry-driven criterion that explains why low-rank factorization often succeeds in large-scale synchronization problems and guides the choice of p in practice, with performance close to information-theoretic limits.

Abstract

The orthogonal group synchronization problem, which focuses on recovering orthogonal group elements from their corrupted pairwise measurements, encompasses examples such as high-dimensional Kuramoto model on general signed networks, $\mathbb{Z}_2$-synchronization, community detection under stochastic block models, and orthogonal Procrustes problem. The semidefinite relaxation (SDR) has proven its power in solving this problem; however, its expensive computational costs impede its widespread practical applications. We consider the Burer-Monteiro factorization approach to the orthogonal group synchronization, an effective and scalable low-rank factorization to solve large scale SDPs. Despite the significant empirical successes of this factorization approach, it is still a challenging task to understand when the nonconvex optimization landscape is benign, i.e., the optimization landscape possesses only one local minimizer, which is also global. In this work, we demonstrate that if the degree of freedom within the factorization exceeds twice the condition number of the ``Laplacian" (certificate matrix) at the global minimizer, the optimization landscape is absent of spurious local minima. Our main theorem is purely algebraic and versatile, and it seamlessly applies to all the aforementioned examples: the nonconvex landscape remains benign under almost identical condition that enables the success of the SDR. Additionally, we illustrate that the Burer-Monteiro factorization is robust to ``monotone adversaries", mirroring the resilience of the SDR. In other words, introducing ``favorable" adversaries into the data will not result in the emergence of new spurious local minimizers.

Local Geometry Determines Global Landscape in Low-rank Factorization for Synchronization

TL;DR

This work addresses orthogonal group synchronization from noisy pairwise measurements and analyzes when the nonconvex BM factorization landscape is free of spurious local minima. It proves a deterministic, algebraic condition linking the required rank p to the condition number of the SDR certificate (Laplacian) at the global minimizer, ensuring benignity for d = 1 and d ≥ 2 under near-SDR regimes. The results extend to robustness against monotone adversaries and apply across Z2 synchronization, SBMs, O(d) synchronization, and generalized Procrustes, delivering near-optimal noise thresholds and broad practical implications for scalable SDP solvers. Overall, the paper provides a unifying, geometry-driven criterion that explains why low-rank factorization often succeeds in large-scale synchronization problems and guides the choice of p in practice, with performance close to information-theoretic limits.

Abstract

The orthogonal group synchronization problem, which focuses on recovering orthogonal group elements from their corrupted pairwise measurements, encompasses examples such as high-dimensional Kuramoto model on general signed networks, -synchronization, community detection under stochastic block models, and orthogonal Procrustes problem. The semidefinite relaxation (SDR) has proven its power in solving this problem; however, its expensive computational costs impede its widespread practical applications. We consider the Burer-Monteiro factorization approach to the orthogonal group synchronization, an effective and scalable low-rank factorization to solve large scale SDPs. Despite the significant empirical successes of this factorization approach, it is still a challenging task to understand when the nonconvex optimization landscape is benign, i.e., the optimization landscape possesses only one local minimizer, which is also global. In this work, we demonstrate that if the degree of freedom within the factorization exceeds twice the condition number of the ``Laplacian" (certificate matrix) at the global minimizer, the optimization landscape is absent of spurious local minima. Our main theorem is purely algebraic and versatile, and it seamlessly applies to all the aforementioned examples: the nonconvex landscape remains benign under almost identical condition that enables the success of the SDR. Additionally, we illustrate that the Burer-Monteiro factorization is robust to ``monotone adversaries", mirroring the resilience of the SDR. In other words, introducing ``favorable" adversaries into the data will not result in the emergence of new spurious local minimizers.
Paper Structure (18 sections, 14 theorems, 137 equations)

This paper contains 18 sections, 14 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Least squares estimation and convex relaxation
  3. Low-rank optimization via Burer-Monteiro factorization
  4. Optimization landscape
  5. High-dimensional Kuramoto model
  6. Monotone adversaries
  7. Main theorems
  8. Theorem for $d=1$
  9. Example: high-dimensional Kuramoto model
  10. Example: $\hbox{{Z}}_2$-synchronization.
  11. Example: community detection under stochastic block model.
  12. Example: robustness against monotone adversaries
  13. Theorem for $d\geq 2$
  14. Example: $\mathop{\mathrm{O}}\nolimits(d)$-synchronization under additive Gaussian noise.
  15. Example: Generalized orthogonal Procrustes problem.
  16. ...and 3 more sections

Key Result

Theorem 2.1

Suppose $\boldsymbol{X}=\boldsymbol{x}\boldsymbol{x}^{\top}$ with $\boldsymbol{x}\in\{\pm 1\}^n$ is a global minimizer to def:sdr and the certificate matrix $\boldsymbol{L}$ has a strictly positive second smallest eigenvalue $\lambda_2(\boldsymbol{L})$, i.e., def:Ld1 holds. The optimization landscap The only local minimizer to def:bm is also global and satisfies $\boldsymbol{S}\boldsymbol{S}^{\top

Theorems & Definitions (20)

  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Theorem 2.6: Robustness against monotone adversaries
  • Corollary 2.7
  • Theorem 2.8
  • Corollary 2.9: Landscape of orthogonal group synchronization
  • Corollary 2.10: Optimization landscape of general orthogonal Procrustes problem
  • ...and 10 more