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This paper studies the quantization of heavy-tailed data in some fundamentalstatistical estimation problems, where the underlying distributions havebounded moments of some order. We propose to truncate and properly dither thedata prior to a uniform quantization. Our major standpoint is that (near)minimax rates of estimation error are achievable merely from the quantized dataproduced by the proposed scheme. In particular, concrete results are worked outfor covariance estimation, compressed sensing, and matrix completion, allagreeing that the quantization only slightly worsens the multiplicative factor.Besides, we study compressed sensing where both covariate (i.e., sensingvector) and response are quantized. Under covariate quantization, although ourrecovery program is non-convex because the covariance matrix estimator lackspositive semi-definiteness, all local minimizers are proved to enjoy nearoptimal error bound. Moreover, by the concentration inequality of productprocess and covering argument, we establish near minimax uniform recoveryguarantee for quantized compressed sensing with heavy-tailed noise.

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