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 → ∞. Our analysis uncovers a phase transition phenomenon governed by the parameter α = L/(σ² log L), where σ² is the variance of the noise. When α > 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 ε approaches σ²/ε for small ε; this is the sample complexity of estimating a signal in additive white Gaussian noise, which does not involve shifts.