Tuning-free one-bit covariance estimation using data-driven dithering

Sjoerd Dirksen, Johannes Maly

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

We consider covariance estimation of any subgaussian distribution from finitely many i.i.d. samples that are quantized to one bit of information per entry. Recent work has shown that a reliable estimator can be constructed if uniformly distributed dithers on [-λ,λ] are used in the one-bit quantizer. This estimator enjoys near-minimax optimal, non-asymptotic error estimates in the operator and Frobenius norms if λ is chosen proportional to the largest variance of the distribution. However, this quantity is not known a-priori, and in practice λ needs to be carefully tuned to achieve good performance. In this work we resolve this problem by introducing a tuning-free variant of this estimator, which replaces λ by a data-driven quantity. We prove that this estimator satisfies the same non-asymptotic error estimates — up to small (logarithmic) losses and a slightly worse probability estimate. We also show that by using refined data-driven dithers that vary per entry of each sample, one can construct an estimator satisfying the same estimation error bound as the sample covariance of the samples before quantization — again up logarithmic losses. Our proofs rely on a new version of the Burkholder-Rosenthal inequalities for matrix martingales, which is expected to be of independent interest.

Original languageEnglish
Pages (from-to)5228-5247
JournalIEEE Transactions on Information Theory
Volume70
Issue number7
Early online date26 Jan 2024
DOIs
Publication statusPublished - Jul 2024

Keywords

  • Covariance estimation
  • Covariance matrices
  • Dithering
  • Estimation error
  • One-bit quantization
  • Quantization (signal)
  • Random variables
  • Reliability
  • Sensors
  • Symmetric matrices

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