Minimax Estimation of Maximum Mean Discrepancy with Radial Kernels
Ilya Tolstikhin · Bharath Sriperumbudur · Bernhard Schölkopf

Wed Dec 7th 06:00 -- 09:30 PM @ Area 5+6+7+8 #58 #None
Maximum Mean Discrepancy (MMD) is a distance on the space of probability measures which has found numerous applications in machine learning and nonparametric testing. This distance is based on the notion of embedding probabilities in a reproducing kernel Hilbert space. In this paper, we present the first known lower bounds for the estimation of MMD based on finite samples. Our lower bounds hold for any radial universal kernel on $\R^d$ and match the existing upper bounds up to constants that depend only on the properties of the kernel. Using these lower bounds, we establish the minimax rate optimality of the empirical estimator and its $U$-statistic variant, which are usually employed in applications.

Author Information

Ilya Tolstikhin (Google, Brain Team, Zurich)
Bharath Sriperumbudur (Penn State University)
Bernhard Schölkopf (MPI for Intelligent Systems)

Bernhard Scholkopf received degrees in mathematics (London) and physics (Tubingen), and a doctorate in computer science from the Technical University Berlin. He has researched at AT&T Bell Labs, at GMD FIRST, Berlin, at the Australian National University, Canberra, and at Microsoft Research Cambridge (UK). In 2001, he was appointed scientific member of the Max Planck Society and director at the MPI for Biological Cybernetics; in 2010 he founded the Max Planck Institute for Intelligent Systems. For further information, see

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