Optimal Quantization via Dynamics

TL;DR

This paper explores geometric quantization of probability measures using stationary processes from dynamical systems, analyzing their effectiveness in approximating optimal quantization errors and comparing different approximation metrics.

Contribution

It introduces a novel geometric approach to quantization via dynamical systems and examines the approximation quality of stationary processes, including special cases like random and Diophantine processes.

Findings

Stationary processes can approximate optimal quantization with quantifiable error bounds.

Different metrics for measuring approximation have distinct advantages and limitations.

The approach bridges dynamical systems theory with quantization problems.

Abstract

Quantization for probability distributions refers broadly to estimating a given probability measure by a discrete probability measure supported by a finite number of points. We consider general geometric approaches to quantization using stationary processes arising in dynamical systems, followed by a discussion of the special cases of stationary processes: random processes and Diophantine processes. We are interested in how close stationary process can be to giving optimal $n$-means and $n^{th}$ optimal mean distortion errors. We also consider different ways of measuring the degree of approximation by quantization, and their advantages and disadvantages in these different contexts.