The Noise-sensitivity phase transition in spectral group synchronization over compact groups

Elad Romanov*, Matan Gavish

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

In Group Synchronization, one attempts to recover a collection of unknown group elements from noisy measurements of their pairwise differences. Several important problems in vision and data analysis reduce to group synchronization over various compact groups. Spectral Group Synchronization is a commonly used, robust algorithm for solving group synchronization problems, which relies on diagonalization of a block matrix whose blocks are matrix representations of the measured pairwise differences. Assuming uniformly distributed measurement errors, we present a rigorous analysis of the accuracy and noise sensitivity of spectral group synchronization algorithms over any compact group. We identify a noise threshold above which the performance of the algorithm completely breaks down. Below the threshold, we calculate an asymptotically exact formula for the accuracy, up to the rounding error, as a function of the noise level. We also provide a consistent risk estimate, allowing practitioners to estimate the method's accuracy from available measurements.

Original languageAmerican English
Pages (from-to)935-970
Number of pages36
JournalApplied and Computational Harmonic Analysis
Volume49
Issue number3
DOIs
StatePublished - Nov 2020

Bibliographical note

Publisher Copyright:
© 2019 Elsevier Inc.

Keywords

  • Bai-Yin theorem
  • Block random matrix
  • Compact group
  • Haar measure
  • Low-rank matrix recovery
  • Spiked model
  • Synchronization
  • Wigner semicircle law

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