MMD Aggregated Two-Sample Test Article Swipe

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Antonin Schrab , Ilmun Kim , Mélisande Albert , Béatrice Laurent , Benjamin Guedj , Arthur Gretton · YOU? · · 2021 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2110.15073

We propose two novel nonparametric two-sample kernel tests based on the\nMaximum Mean Discrepancy (MMD). First, for a fixed kernel, we construct an MMD\ntest using either permutations or a wild bootstrap, two popular numerical\nprocedures to determine the test threshold. We prove that this test controls\nthe probability of type I error non-asymptotically. Hence, it can be used\nreliably even in settings with small sample sizes as it remains\nwell-calibrated, which differs from previous MMD tests which only guarantee\ncorrect test level asymptotically. When the difference in densities lies in a\nSobolev ball, we prove minimax optimality of our MMD test with a specific\nkernel depending on the smoothness parameter of the Sobolev ball. In practice,\nthis parameter is unknown and, hence, the optimal MMD test with this particular\nkernel cannot be used. To overcome this issue, we construct an aggregated test,\ncalled MMDAgg, which is adaptive to the smoothness parameter. The test power is\nmaximised over the collection of kernels used, without requiring held-out data\nfor kernel selection (which results in a loss of test power), or arbitrary\nkernel choices such as the median heuristic. We prove that MMDAgg still\ncontrols the level non-asymptotically, and achieves the minimax rate over\nSobolev balls, up to an iterated logarithmic term. Our guarantees are not\nrestricted to a specific type of kernel, but hold for any product of\none-dimensional translation invariant characteristic kernels. We provide a\nuser-friendly parameter-free implementation of MMDAgg using an adaptive\ncollection of bandwidths. We demonstrate that MMDAgg significantly outperforms\nalternative state-of-the-art MMD-based two-sample tests on synthetic data\nsatisfying the Sobolev smoothness assumption, and that, on real-world image\ndata, MMDAgg closely matches the power of tests leveraging the use of models\nsuch as neural networks.\n

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preprint

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http://arxiv.org/abs/2110.15073

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https://arxiv.org/pdf/2110.15073

OA Status

green

References

38

Related Works

10

OpenAlex ID

https://openalex.org/W3209376323

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OpenAlex ID

https://openalex.org/W3209376323

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DOI

https://doi.org/10.48550/arxiv.2110.15073

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Title

MMD Aggregated Two-Sample Test

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Type

preprint

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Publication year

2021

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Publication date

2021-10-28

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Authors

Antonin Schrab, Ilmun Kim, Mélisande Albert, Béatrice Laurent, Benjamin Guedj, Arthur Gretton

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https://arxiv.org/abs/2110.15073

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https://arxiv.org/pdf/2110.15073

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Yes

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green

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https://arxiv.org/pdf/2110.15073

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Concepts

Test (biology), Sample (material), Statistics, Computer science, Mathematics, Chromatography, Chemistry, Geology, Paleontology

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0

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38

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10

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