Approximation Theorems for High-Dimensional Canonical U-Statistics: Gaussian Chaos and Phase Transition
Abstract
We study simultaneous inference for maxima of canonical order-two U-statistics in high dimension. Degeneracy makes quadratic fluctuations leading, so ordinary Gaussian calibration can fail even after exact variance normalization. We show that the appropriate general target is a joint signed Gaussian quadratic chaos and establish a general approximation result that permits indefinite kernels. The general anti-concentration bound is too crude for high-dimensional inference, and we obtain sharper bounds under additional spectral structure. We also identify a phase transition from a non-Gaussian signed-chaos maximum to its covariance-matched Gaussian counterpart driven by the effective rank. For feasible inference, we propose a Gaussian multiplier bootstrap that avoid estimating eigensystems, and establish its validity. Two applications and extensive numerical simulations further illustrate the scope and practical performance of the proposed framework.
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BibTeX
@misc{cai2026approximation,
title = {{Approximation Theorems for High-Dimensional Canonical U-Statistics: Gaussian Chaos and Phase Transition}},
author = {Leheng Cai and Qirui Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20529},
eprint = {2609.20529},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20529},
}
RIS
TY - UNPB
TI - Approximation Theorems for High-Dimensional Canonical U-Statistics: Gaussian Chaos and Phase Transition
AU - Leheng Cai
AU - Qirui Hu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.20529
UR - https://arxiv.org/abs/2609.20529
AB - We study simultaneous inference for maxima of canonical order-two U-statistics in high dimension. Degeneracy makes quadratic fluctuations leading, so ordinary Gaussian calibration can fail even after exact variance normalization. We show that the appropriate general target is a joint signed Gaussian quadratic chaos and establish a general approximation result that permits indefinite kernels. The general anti-concentration bound is too crude for high-dimensional inference, and we obtain sharper bounds under additional spectral structure. We also identify a phase transition from a non-Gaussian signed-chaos maximum to its covariance-matched Gaussian counterpart driven by the effective rank. For feasible inference, we propose a Gaussian multiplier bootstrap that avoid estimating eigensystems, and establish its validity. Two applications and extensive numerical simulations further illustrate the scope and practical performance of the proposed framework.
ER -