A Parameter-Free Two-Bit Covariance Estimator with Improved Operator Norm Error Rate

TL;DR

This paper introduces a new 2-bit covariance matrix estimator that improves operator norm error rates, removes the need for tuning parameters, and adapts to the effective rank of the covariance matrix, enhancing both theoretical and practical performance.

Contribution

The authors propose a parameter-free 2-bit covariance estimator with adaptive dithering scales, achieving improved operator norm error bounds based on effective rank rather than ambient dimension.

Findings

Achieves operator norm error close to sample covariance with halved dithering scales.

Eliminates tuning parameter dependence by data-driven dithering scales.

Performs well in experiments with Gaussian samples, often outperforming previous estimators.

Abstract

A covariance matrix estimator using two bits per entry was recently developed by Dirksen, Maly and Rauhut [Annals of Statistics, 50(6), pp. 3538-3562]. The estimator achieves near minimax rate for general sub-Gaussian distributions, but also suffers from two downsides: theoretically, there is an essential gap on operator norm error between their estimator and sample covariance when the diagonal of the covariance matrix is dominated by only a few entries; practically, its performance heavily relies on the dithering scale, which needs to be tuned according to some unknown parameters. In this work, we propose a new 2-bit covariance matrix estimator that simultaneously addresses both issues. Unlike the sign quantizer associated with uniform dither in Dirksen et al., we adopt a triangular dither prior to a 2-bit quantizer inspired by the multi-bit uniform quantizer. By employing dithering scales varying across entries, our estimator enjoys an improved operator norm error rate that depends on the effective rank of the underlying covariance matrix rather than the ambient dimension, thus closing the theoretical gap. Moreover, our proposed method eliminates the need of any tuning parameter, as the dithering scales are entirely determined by the data. Experimental results under Gaussian samples are provided to showcase the impressive numerical performance of our estimator. Remarkably, by halving the dithering scales, our estimator oftentimes achieves operator norm errors less than twice of the errors of sample covariance.