This paper studies the quantization of heavy-tailed data in some fundamental statistical estimation problems, where the underlying distributions have bounded moments of some order. We propose to truncate and properly dither the data prior to a uniform quantization. Our major standpoint is that (near) minimax rates of estimation error are achievable merely from the quantized data produced by the proposed scheme. In particular, concrete results are worked out for covariance estimation, compressed sensing, and matrix completion, all agreeing that the quantization only slightly worsens the multiplicative factor. Besides, we study compressed sensing where both covariate (i.e., sensing vector) and response are quantized. Under covariate quantization, although our recovery program is non-convex because the covariance matrix estimator lacks positive semi-definiteness, all local minimizers are proved to enjoy near optimal error bound. Moreover, by the concentration inequality of product process and covering argument, we establish near minimax uniform recovery guarantee for quantized compressed sensing with heavy-tailed noise.

翻译：本文研究了在某些基础统计估计问题中对重尾数据进行量化的问题，其中底层分布具有有界阶矩。我们提出在均匀量化之前对数据进行截断并适当抖动。我们的主要观点是，通过所提出方案产生的量化数据，可以实现(近似)极小极大估计误差速率。具体而言，我们为协方差估计、压缩感知和矩阵补全给出了具体结果，所有这些结果都表明量化仅略微恶化乘法因子。此外，我们研究了协变量（即感知向量）和响应均被量化的压缩感知问题。在协变量量化下，尽管由于协方差矩阵估计量缺乏半正定性，我们的恢复程序是非凸的，但所有局部极小值均被证明具有近最优误差界。此外，利用乘积过程的集中不等式和覆盖论证，我们建立了具有重尾噪声的量化压缩感知的近极小极大统一恢复保证。