mlsanityData to 7 Oct 2026

Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD

arXiv 2607.24235 ↗First posted 27 Jul 2026stat.ML cs.LG math.STcited by 2

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Abstract (arXiv)

Over the past 20 years, kernel discrepancies have been leveraged as a highly powerful tool for quantifying the disagreement of distributions, with numerous successful applications in two-sample, goodness-of-fit, and independence testing, among others. Their fastest estimators are known to converge at a parametric rate---n-1/2---under mild conditions. While this rate is known to be minimax optimal on mathbb Rd under strict assumptions with bounded kernels, little is known about its optimality beyond the finite-dimensional Euclidean setting with unbounded kernels. In this work, we prove that the minimax lower bound of estimation of the most popular kernel discrepancies (maximum mean discrepancy, Hilbert-Schmidt independence criterion and kernel Stein discrepancy; MMD, HSIC, KSD) is n-1/2 on general topological spaces, and under mild assumptions on the kernel; the same rates are shown (as corollaries) to hold for the estimation of the mean embedding and the centered cross-covariance operator. Our results settle the question of optimal estimation of these kernel discrepancies.

Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 80 of its 143 references that are in the map. First posted on arXiv 27 Jul 2026.

Builds on

80 of 143 references.

References · 80

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38 more
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Citing papers per year: 2026: 2.

2 papers.

  1. 2026Unbounded Characteristic and Universal Kernelscited by 0
  2. 2026Conditional Kernel Stein Discrepancycited by 0