Error estimates for deterministic empirical approximations of probability measures

TL;DR

The paper addresses deterministic approximation of a probability measure $\mu \in \mathcal{P}_q(\mathbb{R}^d)$ by a discrete uniform measure with $N$ atoms, and studies the rate at which the Wasserstein error $e_{N;d,p}(\mu)$ decays as $N$ grows. It develops a fully deterministic, multiscale construction based on a dyadic partition to allocate mass and place points, yielding upper bounds of the form $e_{N;d,p}(\mu) \lesssim C \mathcal{M}_q(\mu)^{1/q} \left( N^{-1/d} \vee N^{-1/p+1/q} \right)$, with a logarithmic correction at the critical dimension $d = \frac{qp}{q-p}$, and analogous results for weak $q$-moments. The work shows that these rates improve on prior bounds in the intermediate regime $p \le d \le \frac{qp}{q-p}$ and provides matching lower bounds in many cases, establishing essentially optimal deterministic quantization rates in all dimensions (except some critical cases). It also clarifies limits of uniform-rate optimality across all measures and extends the analysis to measures with weak moments, contributing to deterministic mean-field and optimal transport approximations in high dimensions.

Abstract

The question of approximating an arbitrary probability measure in the Wasserstein distance by a discrete one with uniform weights is considered. Estimates are obtained for the rate of convergence as the number of points tends to infinity, depending on the moment parameter, the parameter in the Wasserstein distance, and the dimension. In certain low-dimensional regimes and for measures with unbounded support, the rates are improvements over those obtained through other methods, including through random sampling. Except for some critical cases, the rates are shown to be optimal.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['thm:optimal_quantize']}
• proof : Proof of Theorem \ref{['thm:optimal_quantize_weak']}
• Proposition 5
• Lemma 6