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Zador Theorem for optimal quantization with respect to Bregman divergences

Guillaume Boutoille, Gilles Pagès

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.

Zador Theorem for optimal quantization with respect to Bregman divergences

Abstract

We establish a Zador like theorem for -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.

Paper Structure

This paper contains 15 sections, 11 theorems, 227 equations, 1 figure.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Definitions and background on $L^r$-optimal quantization w.r.t. a norm
  4. Short background on regular $L^r$-optimal vector quantization
  5. Sharp rate for $L^r$-optimal quantization : Zador's theorem
  6. Optimal vector quantization with respect to a Bregman divergence: definitions and first properties
  7. Bregman divergence
  8. Introduction to quantization with respect to a Bregman divergence
  9. Existence of optimal quantizers (background)
  10. Asymptotic analysis of the quantization error: a Zador like theorem
  11. Zador's like Theorem for Bregman divergence
  12. Proof of Theorem \ref{['thm:ZadorBregman']}: Zador's theorem for Bregman divergence
  13. A first step: Zador's Theorem for Mahalanobis divergence and the uniform distribution over $[0,1]^d$
  14. Proof of Theorem \ref{['thm:ZadorBregman']}
  15. Zador's Theorem for matrix-valued fields of symmetric positive definite matrices

Key Result

Theorem 2.1

Let $r>0$ and let $\|\cdot\|$ denote any norm on $\mathbb{R}^d$. $(a)$ Assume $\int_{\mathbb{R}^d}\|\xi\|^{r+\delta}P(d\xi)< + \infty$ for some $\delta>0$. where $h=\frac{\mathrm{d}P^a}{\mathrm{d}\lambda_d}$ denotes the density of the absolutely continuous part $P^a$ of $P$ with respect to the Lebesgue measure $\lambda_d$ on $\mathbb{R}^d$. Furthermore corresponds to the case $P= U([0,1]^d)$. Wh

Figures (1)

  • Figure 1: Firewall lemma.

Theorems & Definitions (14)

  • Definition 2.1: Quantization error
  • Theorem 2.1: Zador, Graf & Luschgy 2000, Luschgy &Pagès 2023
  • Definition 2.2: Bregman divergence associated to strictly convex function $F$
  • Definition 3.1: Quantization Error w.r.t. Bregman divergences
  • Proposition 3.1: Integrability
  • Proposition 3.2
  • Theorem 3.1: Existence when $r=2$
  • Theorem 3.2: Existence when $r>2$
  • Theorem 4.1: Zador like theorem for Bregman divergence
  • Proposition 5.1
  • ...and 4 more