Quantizing Heavy-tailed Data in Statistical Estimation: (Near) Minimax Rates, Covariate Quantization, and Uniform Recovery

TL;DR

This paper demonstrates that near minimax rates in statistical estimation are achievable with heavy-tailed data through a quantization scheme involving truncation and dithering, with applications to covariance estimation, compressed sensing, and matrix completion.

Contribution

It introduces a quantization method that preserves near minimax estimation rates for heavy-tailed data across multiple statistical problems.

Findings

Quantization slightly increases estimation error factors.

Near minimax rates are achievable with the proposed scheme.

Recovery guarantees hold even with heavy-tailed noise.

Abstract

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.