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Asymptotics of the quantization problem on metric measure spaces

Ata Deniz Aydin

TL;DR

This work develops a unifying framework for quantization on general metric measure spaces by introducing quantization coefficients and the notion of $(p,s)$-quantizability, linking asymptotic quantization errors to Hausdorff densities. It proves lower bounds on the quantization coefficient via density integrals and, under mild conditions, matching upper bounds, thereby extending Zador-type results beyond Euclidean settings. A central contribution is the Zador theorem for countably $m$-rectifiable measures on $\mathbb{R}^d$, establishing existence and value of the asymptotic coefficient and describing asymptotic mass distribution of quantizers; a special 1-rectifiable case is also proved for general Polish spaces. The results rely on Pierce-type nonasymptotic bounds, random quantizer methods, Lipschitz/bi-Lipschitz parametrizations, and geometric measure-theoretic tools, providing a principled route to quantization on complex spaces with fractal or nonsmooth structure. These findings clarify when Zador-type rates hold and how the local dimensionality and density influence optimal sampling in Wasserstein terms, with implications for numerical quantization and related transport problems.

Abstract

The problem of quantization of measures looks for best approximations of probability measures on a metric space by discrete measures supported on $N$ points, where the error of approximation is measured with respect to the Wasserstein distance. Zador's theorem states that, for measures on $\mathbb{R}^d$ or $d$-dimensional Riemannian manifolds satisfying appropriate integrability conditions, the quantization error decays to zero as $N \to \infty$ at the rate $N^{-1/d}$. In this paper, we provide a general treatment of the asymptotics of quantization on metric measure spaces $(X, ν)$. We show that a weaker version of Zador's theorem involving the Hausdorff densities of $ν$ holds also in this general setting. We also prove Zador's theorem in full for appropriate $m$-rectifiable measures on Euclidean space, answering a conjecture by Graf and Luschgy in the affirmative. For both results, the higher integrability conditions of Zador's theorem are replaced with a general notion of $(p,s)$-quantizability, which follows from Pierce-type (non-asymptotic) upper bounds on the quantization error, and we also prove multiple such bounds at the level of metric measure spaces.

Asymptotics of the quantization problem on metric measure spaces

TL;DR

This work develops a unifying framework for quantization on general metric measure spaces by introducing quantization coefficients and the notion of -quantizability, linking asymptotic quantization errors to Hausdorff densities. It proves lower bounds on the quantization coefficient via density integrals and, under mild conditions, matching upper bounds, thereby extending Zador-type results beyond Euclidean settings. A central contribution is the Zador theorem for countably -rectifiable measures on , establishing existence and value of the asymptotic coefficient and describing asymptotic mass distribution of quantizers; a special 1-rectifiable case is also proved for general Polish spaces. The results rely on Pierce-type nonasymptotic bounds, random quantizer methods, Lipschitz/bi-Lipschitz parametrizations, and geometric measure-theoretic tools, providing a principled route to quantization on complex spaces with fractal or nonsmooth structure. These findings clarify when Zador-type rates hold and how the local dimensionality and density influence optimal sampling in Wasserstein terms, with implications for numerical quantization and related transport problems.

Abstract

The problem of quantization of measures looks for best approximations of probability measures on a metric space by discrete measures supported on points, where the error of approximation is measured with respect to the Wasserstein distance. Zador's theorem states that, for measures on or -dimensional Riemannian manifolds satisfying appropriate integrability conditions, the quantization error decays to zero as at the rate . In this paper, we provide a general treatment of the asymptotics of quantization on metric measure spaces . We show that a weaker version of Zador's theorem involving the Hausdorff densities of holds also in this general setting. We also prove Zador's theorem in full for appropriate -rectifiable measures on Euclidean space, answering a conjecture by Graf and Luschgy in the affirmative. For both results, the higher integrability conditions of Zador's theorem are replaced with a general notion of -quantizability, which follows from Pierce-type (non-asymptotic) upper bounds on the quantization error, and we also prove multiple such bounds at the level of metric measure spaces.

Paper Structure

This paper contains 26 sections, 38 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Main definitions and results
  3. Density bounds in metric measure spaces
  4. Sufficient conditions for $(p,s)$-quantizability
  5. Zador's theorem for rectifiable measures
  6. Historical notes
  7. Organization
  8. Preliminary definitions and properties
  9. Properties of the quantization error
  10. Quantization coefficients and $(p,s)$-quantizability
  11. Additivity of quantization coefficients
  12. Density estimates on metric measure spaces
  13. Estimates for uniformly bounded densities
  14. Proof of Theorem \ref{['thm:densbd']}
  15. Zador's theorem for rectifiable measures
  16. ...and 11 more sections

Key Result

Theorem 1.4

Let $(X, \nu)$ be a Polish metric measure space, $p \in [1,\infty)$, $s \in (0,\infty)$, and set $p^\prime := \frac{sp}{s+p}$. Then for any $\mu \in {\mathcal{M}}^p_+(X)$, and if in addition $\mu$ is $(p,s)$-quantizable and $\underline{\vartheta}^{(\nu)}_s > 0$$\mu$-a.e., where $C_2 > C_1 > 0$ are explicit constants depending only on $p$ and $s$, and $\rho$ is the density of the absolutely conti

Theorems & Definitions (97)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Hausdorff density bounds
  • Theorem 1.5: Random quantizer condition
  • Corollary 1.6: Volume growth condition
  • Theorem 1.7: Zador's theorem, rectifiable measures on ${\mathbb R}^d$
  • Theorem 1.8: Zador's theorem, $1$-rectifiable measures
  • Definition 2.1
  • Proposition 2.2: Basic properties of $e_p$
  • ...and 87 more