Sharp Rates of MMD Empirical Estimation with Power Kernels

We establish quantitative rates of convergence for the empirical estimation of probability measures by means of the Maximum Mean Discrepancy (MMD) with power kernel Kq(x,y) = -|x-y|q, q ∈ (0,2). The resulting discrepancy is the classical energy distance mathcal Eq2(μ, ω) = -frac12iintℝd × ℝd |x-y|q , d(μ - ω)(x), d(μ - ω)(y), and we ask how fast the best N-point empirical approximation infμN ∈ PNEq(μN,ω) decays as N → ∞. Given a probability measure ω on ℝd with compact support satisfying an Ahlfors regularity condition of exponent β ∈ (0,d], we prove that the sharp two-sided bound mathcal Eq(μN, ω) asymp N^-frac12left(1 + fracqβright) holds both for the worst-case empirical measure μN (lower bound, holding for every configuration of N points) and for an optimally chosen empirical measure μN (upper bound). This complements the qualitative consistency result of Fornasier and H"utter citefornasier2014consistency, who proved narrow convergence of the minimizers of mathcal Eq2(·, ω) over empirical measures without quantitative rates.