The Nystr\"om method for functional quantization with an application to the fractional Brownian motion

TL;DR

This paper applies the Nyström method to compute optimal quantizers for Gaussian processes, specifically deriving the fractional Brownian motion's quantization via its Karhunen-Loève decomposition and testing a variance reduction algorithm.

Contribution

It introduces a novel application of the Nyström method for functional quantization of Gaussian processes, including fractional Brownian motion, with practical numerical validation.

Findings

Successful derivation of fractional Brownian motion quantization

Effective numerical validation of the stratification variance reduction

Demonstration of the Nyström method's applicability to Gaussian process quantization

Abstract

In this article, the so-called "Nystr\"om method" is tested to compute optimal quantizers of Gaussian processes. In particular, we derive the optimal quantization of the fractional Brownian motion by approximating the first terms of its Karhunen-Lo\`eve decomposition. A numerical test of the "functional stratification" variance reduction algorithm is performed with the fractional Brownian motion.