The high resolution vector quantization problem with Orlicz norm distortion

TL;DR

This paper develops a high-resolution formula for vector quantization under Orlicz norm distortion, characterizing optimal point densities and convergence properties of codebooks in this generalized setting.

Contribution

It introduces a variational framework for optimal point density in Orlicz norm quantization and analyzes the asymptotic behavior of optimal codebooks.

Findings

Optimal point density solves a variational problem involving a function g.

Asymptotically optimal codebooks form a tight sequence of empirical measures.

Convergence results similar to classical quantization are established in most cases.

Abstract

We derive a high-resolution formula for the quantization problem under Orlicz norm distortion. In this setting, the optimal point density solves a variational problem which comprises a function $g:\mathbb{R}_+\to[0,\infty)$ characterizing the quantization complexity of the underlying Orlicz space. Moreover, asymptotically optimal codebooks induce a tight sequence of empirical measures. The set of possible accumulation points is characterized and in most cases it consists of a single element. In that case, we find convergence as in the classical setting.