A Fast Approximate Maximum Likelihood Estimator for Low SNR Multi-Reference Alignment

Motivated by single-particle cryo-electron microscopy, multi-reference alignment (MRA) models the task of recovering an unknown signal from multiple noisy observations corrupted by random rotations. The standard approach for computing the maximum likelihood estimator (MLE) is the expectation-maximization (EM) algorithm; however, it often becomes computationally prohibitive, particularly in low signal-to-noise ratio (SNR) settings. We introduce a fast approximate MLE for MRA over the special orthogonal groups $\mathrm{SO}(2)$ and $\mathrm{SO}(3)$ in the low-SNR regime. A low-SNR Taylor expansion of the likelihood reveals a closed-form, non-iterative approximate MLE. We show that this approach yields a consistent estimator in the low-SNR limit and requires substantially lower computational complexity than both EM and invariant-based alternatives. Numerical experiments generated from molecular volumes show that the proposed method provides a favorable accuracy-runtime trade-off, especially in challenging low-SNR regimes, and can serve as an effective initialization for EM.