Minimax estimation of kernel mean embeddings

Ilya Tolstikhin, Bharath K. Sriperumbudur, Krikamol Muandet

Research output: Contribution to journalArticlepeer-review

29 Scopus citations


In this paper, we study the minimax estimation of the Bochner integral (Equation Presented) also called as the kernel mean embedding, based on random samples drawn i.i.d. from P, where k : X × X → ℝ is a positive definite kernel. Various estimators (including the empirical estimator), θn of µk(P) are studied in the literature wherein all of them satisfy ∥θnk(P)∥Hk = OP(n-1/2) with Hk being the reproducing kernel Hilbert space induced by k. The main contribution of the paper is in showing that the above mentioned rate of n-1/2 is minimax in ∥· ∥Hk and ∥· ∥L2(ℝd)-norms over the class of discrete measures and the class of measures that has an infinitely differentiable density, with k being a continuous translation-invariant kernel on ℝd. The interesting aspect of this result is that the minimax rate is independent of the smoothness of the kernel and the density of P (if it exists).

Original languageEnglish (US)
Pages (from-to)1-47
Number of pages47
JournalJournal of Machine Learning Research
StatePublished - Jul 1 2017

All Science Journal Classification (ASJC) codes

  • Software
  • Artificial Intelligence
  • Control and Systems Engineering
  • Statistics and Probability


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