# Asymptotics of the quantization problem on metric measure spaces

arXiv: https://arxiv.org/abs/2503.18779 · HTML: https://mlsanity.com/paper/2503.18779/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Mar 2025.

## Built on by (most cited first, 1 of 1)

- [Gromov-Wasserstein Quantization and Clustering: Structure, Rates, and Algorithms](https://mlsanity.com/paper/2608.11016/) (2026)

## Abstract (arXiv)

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 ℝd or d-dimensional Riemannian manifolds satisfying appropriate integrability conditions, the quantization error decays to zero as N → ∞ at the rate N-1/d. In this paper, we provide a general treatment of the asymptotics of quantization on metric measure spaces (X, nu). We show that a weaker version of Zador's theorem involving the Hausdorff densities of nu 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.