Symmetry: A general structure in nonparametric regression
Louis Goldwater Christie, John A. D. Aston
The Annals of Statistics · 2025 · Volume 53 · Issue 5
Abstract
In this paper, we present the framework of symmetry in nonparametric regression. This generalises the framework of covariate sparsity, where the regression function f:[0,1]d→R depends only on at most s<d of the covariates, which is a special case of translation symmetry with linear orbits. In general, this extends to other types of functions that capture lower dimensional behavior even when these structures are nonlinear. We show both that known symmetries of regression functions can be exploited to give similarly faster rates, and that unknown symmetries with Lipschitz actions can be estimated sufficiently quickly to adaptively obtain the same rates. This is done by explicit constructions of partial symmetrisation operators that are then applied to usual estimators, and with a two-step M-estimator of the maximal symmetry of the regression function. We also demonstrate the finite sample performance of these estimators on synthetic data.
Keywords
62G08, 62R20, Dimension reduction, invariant models, Nonparametric regression
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