Covariance estimation under one-bit quantization

TL;DR

This paper develops new methods for estimating covariance matrices from highly quantized data, specifically one-bit and two-bit measurements, providing theoretical error bounds and demonstrating near-optimality.

Contribution

It introduces novel estimators for covariance from one-bit and two-bit quantized samples, with non-asymptotic error bounds and minimax lower bounds.

Findings

Accurate covariance estimation is possible from one-bit and two-bit quantized data.

Derived non-asymptotic error bounds for the proposed estimators.

Numerical simulations confirm near-optimality of the bounds.

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. We demonstrate the near-optimality of our error bounds by deriving corresponding (minimax) lower bounds and using numerical simulations.