Motivated by structural biology applications, we study the projected multi-reference alignment (MRA) model, in which an unknown signal is observed through noisy samples, each generated by applying a random cyclic shift followed by a fixed projection. The projection merges reflection-symmetric index pairs, thereby discarding orientation information. The goal is to recover the dihedral orbit of the signal. We prove that in the high-noise regime, the first three moments of the projected observations determine a generic dihedral orbit. The main mechanism is a reduction, at the moment level, from projected MRA to the reflection-invariant phase-coupling structure of dihedral MRA. In Fourier-cosine coordinates adapted to the projection, the first moment determines the mean component, the second moment determines the Fourier magnitudes, and selected third moments yield the cosine phase-coupling relations appearing in the dihedral bispectrum. These relations lead to a constructive recovery scheme from moments up to order three. We complement the population theory with finite-sample experiments comparing expectation--maximization (EM), direct moment optimization, and direct Fourier-cosine moment optimization. The results show that, in the high-noise regime, both EM and direct moment optimization are consistent with the predicted third-moment sample-complexity scaling $n \gtrsim σ^6$, where $n$ is the number of observations and $σ^2$ is the noise variance.

翻译：受结构生物学应用启发，本文研究了投影多参考对齐（MRA）模型，其中未知信号通过含噪观测样本被观测，每个样本由随机循环移位后施加固定投影生成。投影合并反射对称的指标对，从而丢弃方向信息。目标是恢复信号的双面轨道。我们证明，在高噪声条件下，投影观测的前三阶矩可确定一般性双面轨道。主要机制是在矩层面将投影MRA约简为双面MRA的反射不变相位耦合结构。在适应投影的傅里叶-余弦坐标系中，一阶矩确定均值分量，二阶矩确定傅里叶幅度，而选定的三阶矩提供双面双谱中出现的余弦相位耦合关系。这些关系导出了从三阶矩进行重构的构造性方案。我们通过有限样本实验对总体理论进行补充，比较了期望最大化（EM）、直接矩优化和直接傅里叶-余弦矩优化。结果表明，在高噪声条件下，EM和直接矩优化均与预测的三阶矩样本复杂度标度$n \gtrsim σ^6$一致，其中$n$为观测数，$σ^2$为噪声方差。