Expectation-maximization for multi-reference alignment: Two pitfalls and one remedy

Abstract: We study the multi-reference alignment model, which involves recovering a signal from noisy observations that have been randomly transformed by an unknown group action, a fundamental challenge in statistical signal processing, computational imaging, and structural biology. While much of the theoretical literature has focused on the asymptotic sample complexity of this model, the practical performance of reconstruction algorithms, particularly of the omnipresent expectation maximization (EM) algorithm, remains poorly understood. In this work, we present a detailed investigation of EM in the challenging low signal-to-noise ratio (SNR) regime. We identify and characterize two failure modes that emerge in this setting. The first, called Einstein from Noise, reveals a strong sensitivity to initialization, with reconstructions resembling the input template regardless of the true underlying signal. The second phenomenon, referred to as the Ghost of Newton, involves EM initially converging towards the correct solution but later diverging, leading to a loss of reconstruction fidelity. We provide theoretical insights and support our findings through numerical experiments. Finally, we introduce a simple, yet effective modification to EM based on mini-batching, which mitigates the above artifacts. Supported by both theory and experiments, this mini-batching approach processes small data subsets per iteration, reducing initialization bias and computational cost, while maintaining accuracy comparable to full-batch EM.