Multi-reference alignment in high dimensions: sample complexity and phase transition
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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/(σ^2log L), where σ^2 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 σ^2/ε for small ε; this is the sample complexity of estimating a signal in additive white Gaussian noise, which does not involve shifts. In sharp contrast, when α≤ 2, the problem is significantly harder and the sample complexity grows substantially quicker with σ^2.
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