A High-dimensional Convergence Theorem for U-statistics with Applications to Kernel-based Testing

• URL: http://arxiv.org/abs/2302.05686v3
• Date: Sun, 2 Jul 2023 10:23:11 GMT
• Title: A High-dimensional Convergence Theorem for U-statistics with Applications to Kernel-based Testing
• Authors: Kevin H. Huang, Xing Liu, Andrew B. Duncan, Axel Gandy
• Abstract summary: We prove a convergence theorem for U-statistics of degree two, where the data dimension $d$ is allowed to scale with sample size $n$. We apply our theory to two popular kernel-based distribution tests, MMD and KSD, whose high-dimensional performance has been challenging to study.
• Score: 3.469038201881982
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We prove a convergence theorem for U-statistics of degree two, where the data dimension $d$ is allowed to scale with sample size $n$. We find that the limiting distribution of a U-statistic undergoes a phase transition from the non-degenerate Gaussian limit to the degenerate limit, regardless of its degeneracy and depending only on a moment ratio. A surprising consequence is that a non-degenerate U-statistic in high dimensions can have a non-Gaussian limit with a larger variance and asymmetric distribution. Our bounds are valid for any finite $n$ and $d$, independent of individual eigenvalues of the underlying function, and dimension-independent under a mild assumption. As an application, we apply our theory to two popular kernel-based distribution tests, MMD and KSD, whose high-dimensional performance has been challenging to study. In a simple empirical setting, our results correctly predict how the test power at a fixed threshold scales with $d$ and the bandwidth.

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