Zador Theorem for optimal quantization with respect to Bregman divergences

TL;DR

This paper extends Zador's theorem to optimal vector quantization using Bregman divergences, providing rigorous proofs and handling complex matrix-valued cases.

Contribution

It establishes a Zador-like theorem for Bregman divergences and matrix-valued fields, advancing the theoretical understanding of quantization with these measures.

Findings

Proves a Zador-like theorem for twice differentiable Bregman divergences.

Extends results to matrix-valued Bregman divergences with positive definite matrices.

Develops a rigorous proof strategy overcoming specific technical challenges.

Abstract

We establish a Zador like theorem for $L^r$-optimal vector quantization when the similarity measure is a twice differentiable Bregman divergence of a strictly convex function. On our way we also prove a similar result when the Bregman divergence is replaced by a continuous matrix-valued vector field having values in the set of positive definite matrices. We adopt the strategy of the first fully rigorous proof of the original Zador' theorem (when the similarity measure is the power of a norm). We have to overcome several difficulties which are specific to this framework especially concerning the so-called firewall lemma.