Asymptotic quantization for probability measures on Riemannian manifolds

TL;DR

This paper extends the theory of asymptotic quantization of probability measures to Riemannian manifolds, providing error estimates that depend on manifold curvature and demonstrating the sharpness of their assumptions.

Contribution

It generalizes quantization error estimates from Euclidean spaces to curved Riemannian manifolds under curvature-dependent growth conditions.

Findings

Derived asymptotic quantization error estimates on Riemannian manifolds.

Identified curvature-dependent growth conditions for measures.

Provided an example confirming the sharpness of the assumptions.

Abstract

In this paper we study the quantization problem for probability measures on Riemannian manifolds. Under a suitable assumption on the growth at infinity of the measure we find asymptotic estimates for the quantization error, generalizing the results on $\mathbb{R}^d.$ Our growth assumption depends on the curvature of the manifold and reduces, in the flat case, to a moment condition. We also build an example showing that our hypothesis is sharp.