Under Ahlfors regularity of exponent β, the minimal energy distance between a measure and its N-point empirical version decays exactly as N to the power -½(1 + q/β) for power kernels with exponent q in (0,2).
Kernel two-sample tests in high dimensions: interplay between moment discrepancy and dimension-and-sample orders.Biometrika, 110(2):411–430
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Sharp Rates of MMD Empirical Estimation with Power Kernels
Under Ahlfors regularity of exponent β, the minimal energy distance between a measure and its N-point empirical version decays exactly as N to the power -½(1 + q/β) for power kernels with exponent q in (0,2).