Minimax estimation of maximum mean discrepancy with radial kernels

Ilya Tolstikhin, Bharath K. Sriperumbudur, Bernhard Schölkopf

Research output: Contribution to journalConference articlepeer-review

67 Scopus citations


Maximum Mean Discrepancy (MMD) is a distance on the space of probability measures which has found numerous applications in machine learning and nonpara-metric 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 Rd 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.

Original languageEnglish (US)
Pages (from-to)1938-1946
Number of pages9
JournalAdvances in Neural Information Processing Systems
StatePublished - 2016
Event30th Annual Conference on Neural Information Processing Systems, NIPS 2016 - Barcelona, Spain
Duration: Dec 5 2016Dec 10 2016

All Science Journal Classification (ASJC) codes

  • Computer Networks and Communications
  • Information Systems
  • Signal Processing


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