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

TL;DR

This work addresses the multi-reference alignment problem for the continuous group $SO(2)$, proposing two provable, computationally efficient algorithms—Frequency Marching and a Spectral method—that recover a signal orbit from noisy, randomly rotated observations under a nonuniform rotation distribution. By leveraging the first two population moments, the authors establish exact recoveries in ideal moment settings and provide explicit error bounds when moment-based approximations are imperfect, with optimal estimation rates in high-noise regimes. The methods are developed for both 1-D bandlimited signals and 2-D bandlimited images, including detailed analyses and 2-D extensions using Toeplitz and circulant structures, and are validated via numerical experiments that illustrate performance across SNR and sample size. The work advances the goal of provable, scalable algorithms for MRA and paves the way for more rigorous cryo-EM methodologies, addressing the reliability concerns of heuristic approaches in high-stakes structural biology.

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.

Figures (1)

• Figure 1: (a) Recovery error of the frequency marching and spectral algorithms, with 20% error margins, as a function of the SNR for $n = 10^6$ observations. (b) Recovery error of the frequency marching and spectral algorithms, with 20% error margins, as a function of the number of observations for $\text{SNR} = 100$. (c) Recovery error of the spectral algorithm as a function of $S_B(\hat{\rho})$, compared to the theoretical bound from Theorem \ref{['thm:spectral_bound_1D']}.

Theorems & Definitions (23)

• Lemma 2.1
• Proposition 2.2
• Proposition 2.3
• Theorem 2.4
• Lemma 3.1
• Proposition 3.2
• proof
• proof
• proof
• proof : Proof of Theorem \ref{['thm:spectral_bound_1D']}