December 2022 Covariance estimation under one-bit quantization
Sjoerd Dirksen, Johannes Maly, Holger Rauhut
Author Affiliations +
Ann. Statist. 50(6): 3538-3562 (December 2022). DOI: 10.1214/22-AOS2239

Abstract

We consider the classical problem of estimating the covariance matrix of a sub-Gaussian distribution from i.i.d. samples in the novel context of coarse quantization, that is, instead of having full knowledge of the samples, they are quantized to one or two bits per entry. This problem occurs naturally in signal processing applications. We introduce new estimators in two different quantization scenarios and derive nonasymptotic estimation error bounds in terms of the operator norm. In the first scenario, we consider a simple, scale-invariant one-bit quantizer and derive an estimation result for the correlation matrix of a centered Gaussian distribution. In the second scenario, we add random dithering to the quantizer. In this case, we can accurately estimate the full covariance matrix of a general sub-Gaussian distribution by collecting two bits per entry of each sample. In both scenarios, our bounds apply to masked covariance estimation. We demonstrate the near optimality of our error bounds by deriving corresponding (minimax) lower bounds and using numerical simulations.

Funding Statement

The authors were supported by the DFG through the project CoCoMIMO funded within the priority program SPP 1798 Compressed Sensing in Information Processing (COSIP).

Acknowledgments

The authors are very grateful to the Associate Editor and the anonymous reviewer for their detailed comments, which led to several improvements in this work.

Citation

Download Citation

Sjoerd Dirksen. Johannes Maly. Holger Rauhut. "Covariance estimation under one-bit quantization." Ann. Statist. 50 (6) 3538 - 3562, December 2022. https://doi.org/10.1214/22-AOS2239

Information

Received: 1 April 2021; Revised: 1 April 2022; Published: December 2022
First available in Project Euclid: 21 December 2022

MathSciNet: MR4524507
zbMATH: 07641136
Digital Object Identifier: 10.1214/22-AOS2239

Subjects:
Primary: 62H12
Secondary: 62F12

Keywords: Covariance estimation , quantization

Rights: Copyright © 2022 Institute of Mathematical Statistics

Vol.50 • No. 6 • December 2022
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