April 2022 Iterative algorithm for discrete structure recovery
Chao Gao, Anderson Y. Zhang
Author Affiliations +
Ann. Statist. 50(2): 1066-1094 (April 2022). DOI: 10.1214/21-AOS2140


We propose a general modeling and algorithmic framework for discrete structure recovery that can be applied to a wide range of problems. Under this framework, we are able to study the recovery of clustering labels, ranks of players, signs of regression coefficients, cyclic shifts and even group elements from a unified perspective. A simple iterative algorithm is proposed for discrete structure recovery, which generalizes methods including Lloyd’s algorithm and the power method. A linear convergence result for the proposed algorithm is established in this paper under appropriate abstract conditions on stochastic errors and initialization. We illustrate our general theory by applying it on several representative problems: (1) clustering in Gaussian mixture model, (2) approximate ranking, (3) sign recovery in compressed sensing, (4) multireference alignment and (5) group synchronization, and show that minimax rate is achieved in each case.

Funding Statement

Research of Chao Gao is supported in part by NSF CAREER award DMS-1847590 and NSF grant CCF-1934931. Research of Anderson Y. Zhang is supported in part by NSF grant DMS-2112988.


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Chao Gao. Anderson Y. Zhang. "Iterative algorithm for discrete structure recovery." Ann. Statist. 50 (2) 1066 - 1094, April 2022. https://doi.org/10.1214/21-AOS2140


Received: 1 September 2020; Revised: 1 September 2021; Published: April 2022
First available in Project Euclid: 7 April 2022

MathSciNet: MR4404929
zbMATH: 1486.62058
Digital Object Identifier: 10.1214/21-AOS2140

Primary: 62F07

Keywords: approximate ranking , group synchronization , High-dimensional statistics , k-means clustering , multireference alignment

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 2 • April 2022
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