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.
翻译：我们研究从多声观测信号的多振声中估计信号轨道的双向多参考比对齐问题,由异相组群的随机元素进行计算。我们显示,如果组元素从一般分布中抽取,一般信号的轨道从观测的第二个时刻起就被独有地确定。这意味着,高噪音系统中的最佳估计率与噪音差异的平方成正比。这是这种类型的第一个结果,即,对非贝氏组群和非统一分布的组群进行多重参照比对。根据变异理论和代数几何测量工具,我们还为从一般分布中抽取的组群元素时(即将异相组组替换为一般定数组时),定数组群群元素的多参照调整模型(即,正数组组组被一般定数组取代时)的独特轨道恢复条件。最后,我们设计和研究三个计算框架,用于根据群体同步、预期-氧化和瞬时法估算信号。
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