Multireference alignment (MRA) is the task of recovering a hidden "signal" vector, given many noisy copies that have been cyclically shifted by unknown offsets. This task belongs to the class of orbit recovery problems, in which the observed samples are affected by some group action. These problems have a variety of practical motivations, including the reconstruction of 3-dimensional molecular structure from cryogenic electron microscopy (cryo-EM) images. We consider two variants of MRA: dihedral MRA, where the cyclic group is replaced by the dihedral group, allowing for reversals of the vector in addition to shifts; and projected MRA, where the observations are passed through a projection operator akin to the tomographic projection present in cryo-EM. We apply the method of moments and aim to recover the signal from the third moment tensor of the samples. This inverse problem is well understood for basic MRA, but for the variants we consider there is no polynomial-time algorithm known to succeed for generic signals. We give the first such algorithm for both of these variants. Our method requires the signal length to be a power of two, and recursively subdivides the problem into smaller problems of half the size. The algorithm's success for generic signals is proven, conditional on a conjecture about the rank of a certain symbolic matrix of polynomials. For any given problem size, this conjecture can be verified on a computer.

翻译：多参考对齐（MRA）是一项任务，旨在从受到未知偏移量循环移位的大量含噪副本中恢复隐藏的“信号”向量。该任务属于轨道恢复问题类别，其中观测样本受到某种群作用的影响。这些问题具有多种实际应用背景，包括从冷冻电镜（cryo-EM）图像中重建三维分子结构。我们考虑两种MRA变体：二面体MRA，其中循环群被二面体群取代，允许向量在移位之外还可进行反转；以及投影MRA，其中观测值通过一个投影算子（类似于冷冻电镜中的断层投影）处理。我们采用矩方法，旨在从样本的三阶矩张量中恢复信号。这一逆问题在基本MRA中已有充分理解，但对于我们所考虑的变体，尚无已知的多项式时间算法能成功处理一般信号。我们首次为这两种变体提出了这样的算法。该方法要求信号长度为2的幂，并递归地将问题细分为规模减半的更小子问题。在关于某个多项式符号矩阵秩的猜想成立的前提下，我们证明了该算法对一般信号的成功性。对于任意给定问题规模，该猜想可通过计算机验证。