Asymptotic quantization on Riemannian manifolds via covering growth estimates

TL;DR

The paper addresses asymptotics of quantization errors of probability measures on Riemannian manifolds by replacing the local exponential-map growth condition with a global metric growth condition captured by $O(f)$ covering growth around a base point. It develops a Pierce-type upper bound and a Zador-type asymptotic under finite $(p+\delta)$-moments controlled by the covering-growth function, using a radial annulus partition and sphere coverings to construct near-optimal quantizers. Key contributions include a metric-only framework for asymptotics, explicit bounds under nonnegative Ricci curvature yielding $O(R)$ covering growth, and polynomial-growth results for manifolds with geometric group actions, with potential extension to non-smooth metric measure spaces via Bishop-Gromov volume comparison. The results broaden applicability of quantization asymptotics beyond smooth curvature restrictions and provide a scalable, large-scale geometric criterion for quantization on general spaces.

Abstract

The quantization problem looks for best approximations of a probability measure on a given metric space by finitely many points, where the approximation error is measured with respect to the Wasserstein distance. On particular smooth domains, such as $\mathbb{R}^d$ or complete Riemannian manifolds, the quantization error is known to decay polynomially as the number of points is taken to infinity, provided the measure satisfies an integral condition which controls the amount of mass outside compact sets. On Riemannian manifolds, the existing integral condition involves a quantity measuring the growth of the exponential map, for which the only available estimates are in terms of lower bounds on sectional curvature. In this paper, we provide a more general integral condition for the asymptotics of the quantization error on Riemannian manifolds, given in terms of the growth of the covering numbers of spheres, which is purely metric in nature and concerns only the large-scale growth of the manifold. We further estimate the covering growth of manifolds in two particular cases, namely lower bounds on the Ricci curvature and geometric group actions by a discrete group of isometries. These estimates can themselves generalize beyond manifolds, and hint at a future treatment of asymptotic quantization also on non-smooth metric measure spaces.

Figures (8)

• Figure 1: A maximal $r$-packing on a region $A$ with cardinality $16$ yielding a $2r$-cover of $A$.
• Figure 2: Quantization of a non-uniform distribution on ${\mathbb R}^2$ with $N = 26$, obtained by covering 5 concentric circles with 5 points each.
• Figure 3: Mass lying outside a sphere of radius $R$, first projected to the sphere, and then to the nearest element of a cover of the sphere.
• Figure 4: An $r$-packing on the sphere $\partial B_R(x_0)$, contained in the annulus $B_{R+r}(x_0) \setminus B_{R-r}(x_0)$.
• Figure 5: A parabolic $2$-dimensional surface on ${\mathbb R}^3$ with a fold along one direction. The surface is negatively curved along the fold, and the curvature can be made arbitrarily negative by sharpening the fold, causing $\mathop{}\!\mathrm{d} \exp_{x_0}$ to blow up along the same direction in the tangent space at $x_0$ and making ${\mathcal{A}}_{x_0}$ grow exponentially with $R$. However, the perimeters of spheres remain at most linearly proportional to the radius due to the overall parabolic growth of the surface.

Theorems & Definitions (80)

• Lemma 1.1: Pierce pierce, Graf and Luschgy quantbook
• Theorem 1.2: Iacobelli iacasym
• Theorem 1.3: Iacobelli iacasym
• Corollary 1.4: Iacobelli iacasym
• Definition 1.5
• Remark
• Theorem 1.6
• Theorem 1.7
• Proposition 1.8
• Definition 2.1