Dihedral Multi-Reference Alignment

Tamir Bendory, Dan Edidin, William Leeb, Nir Sharon

Research output: Contribution to journalArticlepeer-review

Abstract

We study the dihedral multi-reference alignment problem of estimating the orbit of a signal from multiple noisy observations of the signal, acted on by random elements of the dihedral group. We show that if the group elements are drawn from a generic distribution, the orbit of a generic signal is uniquely determined from the second moment of the observations. This implies that the optimal estimation rate in the high noise regime is proportional to the square of the variance of the noise. This is the first result of this type for multi-reference alignment over a non-abelian group with a non-uniform distribution of group elements. Based on tools from invariant theory and algebraic geometry, we also delineate conditions for unique orbit recovery for multi-reference alignment models over finite groups (namely, when the dihedral group is replaced by a general finite group) when the group elements are drawn from a generic distribution. Finally, we design and study numerically three computational frameworks for estimating the signal based on group synchronization, expectation-maximization, and the method of moments.

Original languageEnglish
Pages (from-to)3489-3499
Number of pages11
JournalIEEE Transactions on Information Theory
Volume68
Issue number5
DOIs
StatePublished - 1 May 2022

Keywords

  • Expectation-maximization
  • Group synchronization
  • Multi-reference alignment
  • The method of moments

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