标题： 多参考对齐的可证明算法 over $\SO(2)$ 显示英文标题

标题： Provable algorithms for multi-reference alignment over $\SO(2)$

摘要： 多参考对齐（MRA）问题涉及从多个带噪声的观测值中重构信号，每个观测值都由一个随机群元素变换得到。本文聚焦于平面内旋转群 \(\mathrm{SO}(2)\)，并提出了两种在群的非均匀分布下具有准确信号恢复理论保证的计算高效算法。第一个算法利用数据二阶矩的谱特性，而第二个算法则利用频率推进原理。这两种算法在高噪声条件下达到了最优估计率，标志着在群上估计问题的计算高效且统计最优方法开发中的重大进展。 显示英文摘要

摘要： The multi-reference alignment (MRA) problem involves reconstructing a signal from multiple noisy observations, each transformed by a random group element. In this paper, we focus on the group \(\mathrm{SO}(2)\) of in-plane rotations and propose two computationally efficient algorithms with theoretical guarantees for accurate signal recovery under a non-uniform distribution over the group. The first algorithm exploits the spectral properties of the second moment of the data, while the second utilizes the frequency marching principle. Both algorithms achieve the optimal estimation rate in high-noise regimes, marking a significant advancement in the development of computationally efficient and statistically optimal methods for estimation problems over groups.