Multi-Reference Alignment in High Dimensions: Sample Complexity and Phase Transition

Multi-reference alignment entails estimating a signal in R-L from its circularly shifted and noisy copies. This problem has been studied thoroughly in recent years, focusing on the finite-dimensional setting (fixed L). Motivated by single-particle cryo-electron microscopy, we analyze the sample complexity of the problem in the high-dimensional regime L -> infinity. Our analysis uncovers a phase transition phenomenon governed by the parameter alpha = L/(sigma(2) log L), where sigma(2) is the variance of the noise. When alpha > 2, the impact of the unknown circular shifts on the sample complexity is minor. Namely, the number of measurements required to achieve a desired accuracy epsilon approaches sigma(2)/epsilon for small epsilon this is the sample complexity of estimating a signal in additive white Gaussian noise, which does not involve shifts. In sharp contrast, when alpha