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SUMMARY:A High-dimensional Convergence Theorem for U-statistics with Appli
 cations to Kernel-based Testing - Kevin Han Huang (Gatsby Unit\, UCL)
DTSTART:20230526T143000Z
DTEND:20230526T160000Z
UID:TALK201034@talks.cam.ac.uk
CONTACT:97804
DESCRIPTION: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 fi
 nd that the limiting distribution of a U-statistic undergoes a phase trans
 ition from the non-degenerate Gaussian limit to the degenerate limit\, reg
 ardless of its degeneracy and depending only on a moment ratio. A surprisi
 ng consequence is that a non-degenerate U-statistic in high dimensions can
  have a non-Gaussian limit with a larger variance and asymmetric distribut
 ion. Our bounds are valid for any finite n and d\, independent of individu
 al eigenvalues of the underlying function\, and dimension-independent unde
 r a mild assumption. As an application\, we apply our theory to two popula
 r kernel-based distribution tests\, MMD and KSD\, whose high-dimensional\n
 performance has been challenging to study. In a simple empirical setting\,
  our results correctly predict how the test power at a fixed threshold sca
 les with d and the bandwidth.
LOCATION:MR12\, Centre for Mathematical Sciences\, Wilberforce Road\, Camb
 ridge
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