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

arXiv: https://arxiv.org/abs/2308.16059 · HTML: https://mlsanity.com/paper/2308.16059/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 40 references that are in the map.
- First posted on arXiv 30 Aug 2023.

## Builds on (most cited first, 9 of 9)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Covariance Estimation: Optimal Dimension-free Guarantees for Adversarial Corruption and Heavy Tails](https://mlsanity.com/paper/2205.08494/) (2022)
- [High Dimensional Statistical Estimation under Uniformly Dithered One-bit Quantization](https://mlsanity.com/paper/2202.13157/) (2022)
- [The Generalized Lasso for Sub-gaussian Measurements with Dithered Quantization](https://mlsanity.com/paper/1807.06976/) (2018)
- [Quantizing Heavy-tailed Data in Statistical Estimation: (Near) Minimax Rates, Covariate Quantization, and Uniform Recovery](https://mlsanity.com/paper/2212.14562/) (2022)
- [Quantizing data for distributed learning](https://mlsanity.com/paper/2012.07913/) (2020)
- [Sub-Gaussian Matrices on Sets: Optimal Tail Dependence and Applications](https://mlsanity.com/paper/2001.10631/) (2020)
- [Quantized Low-Rank Multivariate Regression with Random Dithering](https://mlsanity.com/paper/2302.11197/) (2023)
- [Plug-in Channel Estimation with Dithered Quantized Signals in Spatially Non-Stationary Massive MIMO Systems](https://mlsanity.com/paper/2301.04641/) (2023)

## Built on by (most cited first, 4 of 4)

- [Robust Mean Estimation under Quantization](https://mlsanity.com/paper/2601.07074/) (2026)
- [Bit Efficient Toeplitz Covariance Estimation](https://mlsanity.com/paper/2412.12678/) (2024)
- [Compressive Toeplitz Covariance Estimation From Few-Bit Quantized Measurements With Applications to DOA Estimation](https://mlsanity.com/paper/2512.22527/) (2025)
- [One-Bit Clustering for Two Component Sub-Gaussian Mixture Models](https://mlsanity.com/paper/2606.21873/) (2026)

## Abstract (arXiv)

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.