Optimal Identical Binary Quantizer Design for Distributed Estimation

TL;DR

This paper designs optimal one-bit quantizers for distributed estimation in sensor networks, demonstrating that certain simple quantizers are minimax-optimal under various noise conditions, especially in high-SNR regimes.

Contribution

It introduces a theoretical framework for designing optimal identical binary quantizers using the minimax-CRB criterion, including conditions for simplified probabilistic quantizers and identifying optimal thresholds.

Findings

Threshold-quantizer is minimax-optimal in low-SNR Gaussian noise.

The proposed quantizer outperforms traditional methods in moderate to high-SNR regimes.

Theoretical conditions simplify the design of optimal quantizers.

Abstract

We consider the design of identical one-bit probabilistic quantizers for distributed estimation in sensor networks. We assume the parameter-range to be finite and known and use the maximum Cram\'er-Rao Lower Bound (CRB) over the parameter-range as our performance metric. We restrict our theoretical analysis to the class of antisymmetric quantizers and determine a set of conditions for which the probabilistic quantizer function is greatly simplified. We identify a broad class of noise distributions, which includes Gaussian noise in the low-SNR regime, for which the often used threshold-quantizer is found to be minimax-optimal. Aided with theoretical results, we formulate an optimization problem to obtain the optimum minimax-CRB quantizer. For a wide range of noise distributions, we demonstrate the superior performance of the new quantizer - particularly in the moderate to high-SNR regime.