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仅量化为一比特信息的有限独立同分布样本中，估计任意次高斯分布的协方差问题。近期研究表明，若在单比特量化器中使用在$[-\lambda,\lambda]$上均匀分布的抖动，则可构造出可靠估计量。当$\lambda$与分布的最大方差成正比时，该估计量在算子范数和Frobenius范数下具有近乎极小值最优的非渐近误差界。然而，该参数无法先验获知，实践中需仔细调优$\lambda$才能实现良好性能。本文通过引入该估计量的无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）损失及略差的概率估计。本证明依赖于新版本的矩阵鞅Burkholder-Rosenthal不等式，该不等式预期具有独立研究价值。