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 language | English (US) |
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Pages (from-to) | 3489-3499 |
Number of pages | 11 |
Journal | IEEE Transactions on Information Theory |
Volume | 68 |
Issue number | 5 |
DOIs | |
State | Published - May 1 2022 |
Bibliographical note
Funding Information:The work of Tamir Bendory and Nir Sharon was supported in part by NSF-BSF under Award 2019752. The work of Tamir Bendory was supported in part by Israel Science Foundation (ISF) under Grant 1924/21. The work of Dan Edidin was supported by Simons Collaboration under Grant 708560. The work of William Leeb was supported in part by NSF under Award IIS-1837992.
Publisher Copyright:
© 1963-2012 IEEE.
Keywords
- Expectation-maximization
- Group synchronization
- Multi-reference alignment
- The method of moments