A high-dimensional convergence theorem for U-statistics with applications to kernel-based testing
File(s)
Published version
Author(s)
Huang, KH
Liu, X
Duncan, AB
Gandy, A
Type
Conference Paper
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.
Date Issued
2023
Date Acceptance
2023-07-12
Citation
Proceedings of Machine Learning Research, 2023, 195, pp.3827-3918
ISSN
2640-3498
Publisher
MLResearchPress
Start Page
3827
End Page
3918
Journal / Book Title
Proceedings of Machine Learning Research
Volume
195
Copyright Statement
© 2023 K.H. Huang, X. Liu, A.B. Duncan & A. Gandy.
Identifier
https://proceedings.mlr.press/v195/huang23a.html
Source
The Thirty Sixth Annual Conference on Learning Theory
Publication Status
Published
Start Date
2023-07-12
Finish Date
2023-07-15
Coverage Spatial
Bangalore, India
Date Publish Online
2023