Asymptotically optimal quantization schemes for Gaussian processes

TL;DR

This paper develops asymptotically optimal quantization schemes for Gaussian processes, leveraging eigenvalue regular variation and eigenvector knowledge to transform the problem into finite-dimensional quantization, with applications to Riemann-Liouville processes.

Contribution

It introduces a new class of asymptotically optimal functional quantizers for Gaussian processes based on eigenstructure analysis.

Findings

Achieves order optimal quantization rates for Gaussian processes.

Provides a high-resolution formula for Riemann-Liouville process quantization errors.

Demonstrates the importance of eigenvalue regular variation in quantizer design.

Abstract

We describe quantization designs which lead to asymptotically and order optimal functional quantizers. Regular variation of the eigenvalues of the covariance operator plays a crucial role to achieve these rates. For the development of a constructive quantization scheme we rely on the knowledge of the eigenvectors of the covariance operator in order to transform the problem into a finite dimensional quantization problem of normal distributions. Furthermore we derive a high-resolution formula for the $L^2$-quantization errors of Riemann-Liouville processes.