期刊
IEEE TRANSACTIONS ON INFORMATION THEORY
卷 68, 期 5, 页码 3489-3499出版社
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TIT.2022.3146488
关键词
Orbits; Estimation; Noise level; Complexity theory; Signal to noise ratio; Probability distribution; Discrete Fourier transforms; Multi-reference alignment; the method of moments; group synchronization; expectation-maximization
资金
- United States - Israel Binational Science Foundation (BSF) [2018230]
- BSF [2020159]
- NSF-BSF [2019752]
- Israel Science Foundation (ISF) [1924/21]
- Simons Collaboration [708560]
- NSF [IIS-1837992]
We study the problem of dihedral multi-reference alignment and show that the optimal estimation rate in high noise regime is proportional to the square of the variance of the noise. We also propose several computational frameworks for signal estimation.
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.
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