TY - JOUR
T1 - Dihedral Multi-Reference Alignment
AU - Bendory, Tamir
AU - Edidin, Dan
AU - Leeb, William
AU - Sharon, Nir
N1 - Publisher Copyright:
© 1963-2012 IEEE.
PY - 2022/5/1
Y1 - 2022/5/1
N2 - 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.
AB - 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.
KW - Expectation-maximization
KW - Group synchronization
KW - Multi-reference alignment
KW - The method of moments
UR - http://www.scopus.com/inward/record.url?scp=85123686310&partnerID=8YFLogxK
U2 - 10.1109/TIT.2022.3146488
DO - 10.1109/TIT.2022.3146488
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AN - SCOPUS:85123686310
SN - 0018-9448
VL - 68
SP - 3489
EP - 3499
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 5
ER -