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Title: MMD Aggregated Two-Sample Test
We propose a novel nonparametric two-sample test based on the Maximum Mean Discrepancy (MMD), which is constructed by aggregating tests with different kernel bandwidths. This aggregation procedure, called MMDAgg, ensures that test power is maximised over the collection of kernels used, without requiring held-out data for kernel selection (which results in a loss of test power), or arbitrary kernel choices such as the median heuristic. We work in the non-asymptotic framework, and prove that our aggregated test is minimax adaptive over Sobolev balls. Our guarantees are not restricted to a specific kernel, but hold for any product of one-dimensional translation invariant characteristic kernels which are absolutely and square integrable. Moreover, our results apply for popular numerical procedures to determine the test threshold, namely permutations and the wild bootstrap. Through numerical experiments on both synthetic and real-world datasets, we demonstrate that MMDAgg outperforms alternative state-of-the-art approaches to MMD kernel adaptation for two-sample testing.
Antonin Schrab is a PhD student at University College London who is jointly supervised by Benjamin Guedj at the UCL AI Centre and Arthur Gretton at the Gatsby Computational Neuroscience Unit. His research interests include kernel methods, PAC-Bayes and generative models. He has recently focused on proving theoretical guarantees for kernel-based aggregated testing procedures.
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