In this paper, we propose a uniformly dithered one-bit quantization scheme for high-dimensional statistical estimation. The scheme contains truncation, dithering, and quantization as typical steps. As canonical examples, the quantization scheme is applied to three estimation problems: sparse covariance matrix estimation, sparse linear regression, and matrix completion. We study both sub-Gaussian and heavy-tailed regimes, with the underlying distribution of heavy-tailed data assumed to possess bounded second or fourth moment. For each model we propose new estimators based on one-bit quantized data. In sub-Gaussian regime, our estimators achieve optimal minimax rates up to logarithmic factors, which indicates that our quantization scheme nearly introduces no additional cost. In heavy-tailed regime, while the rates of our estimators become essentially slower, these results are either the first ones in such one-bit quantized and heavy-tailed setting, or exhibit some advantages over existing comparable results. More specifically, our results for one-bit compressed sensing feature generality of sensing vector (sub-Gaussian or even heavy-tailed) and tractable convex programming. A novel setting where both measurement and covariate are quantized is also first proposed and studied. For one-bit matrix completion, our method is essentially different from the standard likelihood approach and can handle pre-quantization random noise with unknown distribution. Experimental results on synthetic data are presented to support our theoretical analysis.
翻译:在本文中,我们为高维统计估计建议了一个一致的差幅一位数的量化办法。这个办法包含一个单位数的精确度、抖动和量化的典型步骤。作为典型步骤的典型步骤。作为典型的例子,这个量化办法适用于三个估计问题:共差矩阵估计少,线性回归少,以及矩阵完成。我们研究的是英国次和重尾两种制度,假设重尾数据在第二或第四时刻具有约束性。对于每一个模型,我们根据一位数的定量数据提出新的估算。在亚加西制度下,我们的估算数在对数因素上达到最佳的微缩速率,这表明我们的量化办法几乎没有增加成本。在重尾数体系中,我们的估算率基本变慢了,但这些结果要么是一位四分位数和重尾数的设定,要么是现有可比结果的。更具体地说,我们关于一位数级数的精确度分布率分析结果,一个比位数的精确度分析结果也基本是A级算法的重度研究。