Rates of estimation for highdimensional multireference alignment
Abstract
We study the continuous multireference alignment model of estimating a periodic function on the circle from noisy and circularlyrotated observations. Motivated by analogous highdimensional problems that arise in cryoelectron microscopy, we establish minimax rates for estimating generic signals that are explicit in the dimension $K$. In a highnoise regime with noise variance $\sigma^2 \gtrsim K$, the rate scales as $\sigma^6$ 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 lownoise regime where $\sigma^2 \lesssim K/\log K$, the rate scales instead as $K\sigma^2$, 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 arXiv:
 arXiv:2205.01847
 Bibcode:
 2022arXiv220501847D
 Keywords:

 Mathematics  Statistics Theory;
 Statistics  Applications
 EPrint:
 54 pages