## 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 language | English |
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Pages (from-to) | 5228-5247 |

Journal | IEEE Transactions on Information Theory |

Volume | 70 |

Issue number | 7 |

Early online date | 26 Jan 2024 |

DOIs | |

Publication status | Published - Jul 2024 |

## Keywords

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