Quantization of Gaussian measures with R\'enyi-\alpha-entropy constraints

TL;DR

This paper investigates the optimal quantization of Gaussian measures under Re9nyi-b5-entropy constraints, deriving asymptotic errors and highlighting differences from classical cases.

Contribution

It provides the first sharp asymptotic analysis of quantization errors for Gaussian measures with Re9nyi-b5-entropy constraints for b1 > 1.

Findings

Derived sharp asymptotics for b5 > 1

Connected quantization error to small ball probabilities

Revealed different asymptotic order for b1 > 1

Abstract

We consider the optimal quantization problem with R\'enyi-$\alpha$-entropy constraints for centered Gaussian measures on a separable Banach space. For $\alpha = \infty$ we can compute the optimal quantization error by a moment on a ball. For $\alpha \in {}]1,\infty]$ and large entropy bound we derive sharp asymptotics for the optimal quantization error in terms of the small ball probability of the Gaussian measure. We apply our results to several classes of Gaussian measures. The asymptotical order of the optimal quantization error for $\alpha > 1$ is different from the well-known cases $\alpha = 0$ and $\alpha = 1$.