This paper formulates a general cross validation framework for signal denoising. The general framework is then applied to nonparametric regression methods such as Trend Filtering and Dyadic CART. The resulting cross validated versions are then shown to attain nearly the same rates of convergence as are known for the optimally tuned analogues. There did not exist any previous theoretical analyses of cross validated versions of Trend Filtering or Dyadic CART. To illustrate the generality of the framework we also propose and study cross validated versions of two fundamental estimators; lasso for high dimensional linear regression and singular value thresholding for matrix estimation. Our general framework is inspired by the ideas in Chatterjee and Jafarov (2015) and is potentially applicable to a wide range of estimation methods which use tuning parameters.

翻译：本文提出了一种通用的信号去噪交叉验证框架。该框架随后被应用于非参数回归方法，如趋势滤波和二元CART。所得到的交叉验证版本在收敛速率上几乎与已知的最优调参方法相当。此前尚未存在关于趋势滤波或二元CART交叉验证版本的任何理论分析。为说明该框架的通用性，我们还提出并研究了两种基础估计量的交叉验证版本：高维线性回归的lasso方法以及矩阵估计的奇异值阈值方法。我们的通用框架受Chatterjee和Jafarov（2015）的启发，并可能适用于使用调参参数的大范围估计方法。