Symmetry: A general structure in nonparametric regression
本文提出非参数回归中对称性的框架，将协变量稀疏性推广到更一般的低维结构，并证明已知和未知对称性均可用于提升估计速度，通过构造对称化算子实现自适应估计。
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.