Scalar Quantization with Random Thresholds

TL;DR

This paper analyzes the performance of scalar quantizers with randomly chosen thresholds, showing their distortion and entropy characteristics compared to optimal quantizers, especially at high rates.

Contribution

It provides theoretical bounds on distortion and entropy for random threshold scalar quantizers, highlighting their performance relative to optimal designs.

Findings

Distortion is at most 6 times that of optimal quantizers.

Output entropy decreases by approximately (1-gamma)/ln 2 bits with many levels.

High-rate asymptotic distortion is within a constant factor of the optimal.

Abstract

The distortion-rate performance of certain randomly-designed scalar quantizers is determined. The central results are the mean-squared error distortion and output entropy for quantizing a uniform random variable with thresholds drawn independently from a uniform distribution. The distortion is at most 6 times that of an optimal (deterministically-designed) quantizer, and for a large number of levels the output entropy is reduced by approximately (1-gamma)/(ln 2) bits, where gamma is the Euler-Mascheroni constant. This shows that the high-rate asymptotic distortion of these quantizers in an entropy-constrained context is worse than the optimal quantizer by at most a factor of 6 exp(-2(1-gamma)).