Tuning-Free One-Bit Covariance Estimation Using Data-Driven Dithering

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
$[-\lambda,\lambda]$
are used in the one-bit quantizer. This estimator enjoys near-minimax optimal, non-asymptotic error estimates in the operator and Frobenius norms if
$\lambda $
is chosen proportional to the largest variance of the distribution. However, this quantity is not known a-priori, and in practice
$\lambda $
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
$\lambda $
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 to 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.