Quantized Estimation of Gaussian Sequence Models in Euclidean Balls

TL;DR

This paper extends Pinsker's theorem to scenarios with storage or communication constraints, providing sharp bounds on the tradeoff between encoding bits and estimation risk in Gaussian sequence models.

Contribution

It introduces a quantized estimation framework under bit constraints, establishing Pareto-optimal minimax bounds for Euclidean ball signals.

Findings

Sharp upper and lower bounds for risk under bit constraints

Optimal tradeoff between storage and estimation accuracy

Extension of Pinsker's theorem to constrained encoding scenarios

Abstract

A central result in statistical theory is Pinsker's theorem, which characterizes the minimax rate in the normal means model of nonparametric estimation. In this paper, we present an extension to Pinsker's theorem where estimation is carried out under storage or communication constraints. In particular, we place limits on the number of bits used to encode an estimator, and analyze the excess risk in terms of this constraint, the signal size, and the noise level. We give sharp upper and lower bounds for the case of a Euclidean ball, which establishes the Pareto-optimal minimax tradeoff between storage and risk in this setting.