U-centering as subset ANOVA: edge regression and higher-order theory
Abstract
Unbiased squared distance covariance and the Hilbert-Schmidt independence criterion (HSIC) are fourth-order U-statistics, yet U-centering evaluates them from pairwise arrays in O(n2) operations. We show that U-centering is exactly least-squares residualization after fitting additive endpoint effects. This analysis of variance (ANOVA) interpretation explains the zero row sums and the normalization through residual degrees of freedom. The same geometry gives regression representations of dependence and two-sample statistics, exact permutation variances, and relations among pooled and regression mean squares. For arrays indexed by r-element subsets and n≥2r, higher-order U-centering removes all lower-order subset effects and projects onto a space of dimension binomnr−binomnr−1. For fixed symmetric square-integrable kernels evaluated on independent and identically distributed (i.i.d.) paired observations, we derive the residual covariance matrix, exact mean-squared errors for estimating the highest-order Hoeffding components, unbiased positive semidefinite (PSD) covariance estimates, and moment formulas under independence and permutation. The normalized residual inner product equals a direct unbiased estimator involving up to 2r observations, but can be evaluated in O(nr) operations for fixed r. For four-argument matching kernels, three matrix products yield exactly unbiased estimators of a fourth spectral moment of kernel cross-covariance in O(n3) arithmetic and O(n2) storage. A numerical illustration distinguishes dependence structures with equal HSIC and different spectral concentration.
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BibTeX
@misc{zhang2026ucentering,
title = {{U-centering as subset ANOVA: edge regression and higher-order theory}},
author = {Xianyang Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.01364},
eprint = {2608.01364},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.01364},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - U-centering as subset ANOVA: edge regression and higher-order theory
AU - Xianyang Zhang
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2608.01364
UR - https://arxiv.org/abs/2608.01364
AB - Unbiased squared distance covariance and the Hilbert-Schmidt independence criterion (HSIC) are fourth-order U-statistics, yet U-centering evaluates them from pairwise arrays in O(n^2) operations. We show that U-centering is exactly least-squares residualization after fitting additive endpoint effects. This analysis of variance (ANOVA) interpretation explains the zero row sums and the normalization through residual degrees of freedom. The same geometry gives regression representations of dependence and two-sample statistics, exact permutation variances, and relations among pooled and regression mean squares. For arrays indexed by r-element subsets and n≥2r, higher-order U-centering removes all lower-order subset effects and projects onto a space of dimension binomnr−binomnr−1. For fixed symmetric square-integrable kernels evaluated on independent and identically distributed (i.i.d.) paired observations, we derive the residual covariance matrix, exact mean-squared errors for estimating the highest-order Hoeffding components, unbiased positive semidefinite (PSD) covariance estimates, and moment formulas under independence and permutation. The normalized residual inner product equals a direct unbiased estimator involving up to 2r observations, but can be evaluated in O(n^r) operations for fixed r. For four-argument matching kernels, three matrix products yield exactly unbiased estimators of a fourth spectral moment of kernel cross-covariance in O(n^3) arithmetic and O(n^2) storage. A numerical illustration distinguishes dependence structures with equal HSIC and different spectral concentration.
ER -