Rates of estimation for high-dimensional multireference alignment

Abstract

We study the continuous multireference alignment model of estimating a periodic function on the circle from noisy and circularly-rotated observations. Motivated by analogous high-dimensional problems that arise in cryo-electron microscopy, we establish minimax rates for estimating generic signals that are explicit in the dimension K. In a high-noise regime with noise variance ${\mathit{\sigma }}^2\gtrsim \mathit{K}$, for signals with Fourier coefficients of roughly uniform magnitude, the rate scales as ${\mathit{\sigma }}^6$ and has no further dependence on the dimension. This rate is achieved by a bispectrum inversion procedure, and our analyses provide new stability bounds for bispectrum inversion that may be of independent interest. In a low-noise regime where ${\mathit{\sigma }}^2\lesssim \mathit{K}/log\mathit{K}$, the rate scales instead as $\mathit{K}{\mathit{\sigma }}^2$, and we establish this rate by a sharp analysis of the maximum likelihood estimator that marginalizes over latent rotations. A complementary lower bound that interpolates between these two regimes is obtained using Assouad’s hypercube lemma. We extend these analyses also to signals whose Fourier coefficients have a slow power law decay.