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. Our proof relies on a new version of the Burkholder-Rosenthal inequalities for matrix martingales, which is expected to be of independent interest.

翻译：我们考虑从每个条目量化为一比特信息的有限个独立同分布样本中，对任意次高斯分布进行协方差估计。近期研究表明，若在单比特量化器中采用$[-\lambda,\lambda]$上均匀分布的抖动，可构建可靠的估计量。当$\lambda$选为与分布最大方差成比例时，该估计量在算子范数和Frobenius范数下具有接近极小极大最优的非渐近误差界。然而，该参数无法预先获知，在实际应用中需要精心调参$\lambda$以获得良好性能。本文通过引入该估计量的免调参变体解决此问题，该变体以数据驱动量替代$\lambda$。我们证明该估计量满足相同的非渐近误差界——仅存在微小（对数级）损失和略低的概率估计精度。证明过程依赖于矩阵鞅的Burkholder-Rosenthal不等式新版本，该结果本身具有独立的理论价值。