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

Abstract: 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.