In this paper we present the framework of symmetry in nonparametric regression. This generalises the framework of covariate sparsity, where the regression function 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 non-linear. 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 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.

翻译：本文提出了非参数回归中的对称性框架。该框架推广了协变量稀疏性框架（即回归函数仅依赖于至多$s < d$个协变量的情形），后者是线性轨道平移对称性的特例。一般而言，该框架可延伸至其他能够捕捉低维行为（即使这些结构是非线性的）的函数类型。我们证明：既可利用回归函数的已知对称性实现更快的收敛速率，也可对具有Lipschitz作用的未知对称性进行充分快速的估计，从而获得相同速率。这一成果通过两类方法实现：一是对常用估计量显式构造部分对称化算子，二是采用回归函数最大对称性的两步M估计量。此外，我们还在合成数据上展示了这些估计量的有限样本性能。