Entropy Density and Mismatch in High-Rate Scalar Quantization with Renyi Entropy Constraint

TL;DR

This paper investigates the properties of high-rate scalar quantization under Re9nyi entropy constraints, introducing the concepts of entropy density and mismatch effects, extending classical fixed-rate and variable-rate quantization results.

Contribution

It introduces the concept of entropy density for Re9nyi entropy-constrained quantization and extends mismatch analysis to general entropy order b1, broadening classical results.

Findings

Identifies the entropy density in high-rate quantization

Quantifies distortion contribution of source intervals

Analyzes distortion loss due to source density mismatch

Abstract

Properties of scalar quantization with $r$th power distortion and constrained R\'enyi entropy of order $\alpha\in (0,1)$ are investigated. For an asymptotically (high-rate) optimal sequence of quantizers, the contribution to the R\'enyi entropy due to source values in a fixed interval is identified in terms of the "entropy density" of the quantizer sequence. This extends results related to the well-known point density concept in optimal fixed-rate quantization. A dual of the entropy density result quantifies the distortion contribution of a given interval to the overall distortion. The distortion loss resulting from a mismatch of source densities in the design of an asymptotically optimal sequence of quantizers is also determined. This extends Bucklew's fixed-rate ($\alpha=0$) and Gray \emph{et al.}'s variable-rate ($\alpha=1$) mismatch results to general values of the entropy order parameter $\alpha$.