On MMSE estimation from quantized observations in the nonasymptotic regime

TL;DR

This paper derives nonasymptotic bounds on the MMSE regret caused by quantization in the estimation of scalar and vector random variables from quantized noisy observations, considering independence and concentration properties.

Contribution

It provides the first nonasymptotic bounds on MMSE regret for quantized observations in both scalar and vector estimation settings, including dependence and concentration assumptions.

Findings

Nonasymptotic bounds on MMSE regret for scalar estimation.

Nonasymptotic bounds for vector estimation with subgaussian concentration.

Analysis applicable to dependent observations.

Abstract

This paper studies MMSE estimation on the basis of quantized noisy observations. It presents nonasymptotic bounds on MMSE regret due to quantization for two settings: (1) estimation of a scalar random variable given a quantized vector of $n$ conditionally independent observations, and (2) estimation of a $p$-dimensional random vector given a quantized vector of $n$ observations (not necessarily independent) when the full MMSE estimator has a subgaussian concentration property.