  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.
