Open Access
February 2024 Rates of estimation for high-dimensional multireference alignment
Zehao Dou, Zhou Fan, Harrison H. Zhou
Author Affiliations +
Ann. Statist. 52(1): 261-284 (February 2024). DOI: 10.1214/23-AOS2346


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 σ2K, for signals with Fourier coefficients of roughly uniform magnitude, the rate scales as σ6 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 σ2K/logK, the rate scales instead as Kσ2, 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.

Funding Statement

Z. Fan is supported in part by NSF Grants DMS-1916198, DMS-2142476.
H. H. Zhou is supported in part by NSF Grants DMS-2112918, DMS-1918925 and NIH Grant 1P50MH115716.


The authors would like to thank Yihong Wu for a helpful discussion about KL-divergence in mixture models.


Download Citation

Zehao Dou. Zhou Fan. Harrison H. Zhou. "Rates of estimation for high-dimensional multireference alignment." Ann. Statist. 52 (1) 261 - 284, February 2024.


Received: 1 August 2023; Published: February 2024
First available in Project Euclid: 7 March 2024

MathSciNet: MR4718415
Digital Object Identifier: 10.1214/23-AOS2346

Primary: 62G05
Secondary: 62C20

Keywords: bispectrum inversion , Function estimation , group invariance

Rights: Copyright © 2024 Institute of Mathematical Statistics

Vol.52 • No. 1 • February 2024
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