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 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.

翻译：摘要：本文考虑从有限个独立同分布样本中估计任意次高斯分布的协方差，其中每个条目量化为一比特信息。近期的研究表明，若在一比特量化器中使用$[-\lambda,\lambda]$上的均匀分布抖振，则可构建可靠估计器。当$\lambda$选取与分布最大方差成比例时，该估计量在算子范数与Frobenius范数下享有接近极小极大最优的非渐近误差界。然而，该先验量未知，实际中需精细调节$\lambda$以达良好性能。本文通过引入该估计量的无需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——仅损失微小对数项及略微放宽的概率估计。进一步表明，通过采用随每个样本条目变化的精细化数据驱动抖振，可构建满足与量化前样本协方差相同估计误差界的估计量——同样仅损失对数项。我们的证明依赖于矩阵鞅的Burkholder-Rosenthal不等式新版本，该结果预期具有独立价值。