Sharp Rates of MMD Empirical Estimation with Power Kernels

TL;DR

This paper establishes sharp convergence rates for empirical measures approximating probability measures using the Maximum Mean Discrepancy with power kernels, specifically relating to the energy distance and regularity conditions.

Contribution

It provides the first quantitative, sharp two-sided bounds on the convergence rates of empirical measures in energy distance under Ahlfors regularity.

Findings

Convergence rate of N^{-(1 + q/β)/2} for empirical approximation.

Bounds hold both for worst-case and optimally chosen empirical measures.

Results complement previous qualitative convergence results by providing explicit rates.

Abstract

We establish quantitative rates of convergence for the empirical estimation of probability measures by means of the Maximum Mean Discrepancy (MMD) with power kernel $K_q(x,y) = -|x-y|^q$, $q \in (0,2)$. The resulting discrepancy is the classical energy distance $$\mathcal E_q^2(\mu, \omega) = -\frac{1}{2}\iint_{\mathbb{R}^d \times \mathbb{R}^d} |x-y|^q \, d(\mu - \omega)(x)\, d(\mu - \omega)(y),$$ and we ask how fast the best $N$-point empirical approximation $\inf_{\mu_N \in \mathcal{P}^N}\mathcal{E}_q(\mu_N,\omega)$ decays as $N \to \infty$. Given a probability measure $\omega$ on $\mathbb{R}^d$ satisfying an Ahlfors regularity condition of exponent $\beta$, we prove that the sharp two-sided bound $$\mathcal E_q(\mu_N, \omega) \asymp N^{-\frac{1}{2}\left(1 + \frac{q}{\beta}\right)}$$ holds both for the worst-case empirical measure $\mu_N$ (lower bound, holding for every configuration of $N$ points) and for an optimally chosen empirical measure $\mu_N$ (upper bound). This complements the qualitative consistency result of Fornasier and H\"utter \cite{fornasier2014consistency}, who proved narrow convergence of the minimizers of $\mathcal E_q^2(\cdot, \omega)$ over empirical measures without quantitative rates.