Covariance Estimation under One‐bit Quantization

PAMM, vol. 21

Abstract

Abstract We consider the classical problem of estimating the covariance matrix of a subgaussian distribution from i.i.d. samples in the novel context of coarse quantization, i.e., 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 non‐asymptotic 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 subgaussian distribution by collecting two bits per entry of each sample. In both scenarios, our bounds apply to masked covariance estimation as well. We demonstrate the near‐optimality of our error bounds by deriving corresponding (minimax) lower bounds and using numerical simulations.

Authors 3

1. Sjoerd Dirksen

   Utrecht University

   Affiliation as printed

   Mathematical Institute Utrecht University Budapestlaan 6 NL-3584 CD Utrecht

   Mathematical Institute Utrecht University Budapestlaan 6 NL-3584 CD Utrecht

2. Johannes Maly corresponding

   Catholic University of Eichstätt-Ingolstadt

   Affiliation as printed

   Lehrstuhl für wissenschaftliches Rechnen KU Eichstätt/Ingolstadt Ostenstraße 26 DE-85072 Eichstätt

3. Holger Rauhut Aachen Lehrstuhl für Mathematik der Informationsverarbeitung

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl für Mathematik der Informationsverarbeitung RWTH Aachen Pontdriesch 10 DE-52062 Aachen