Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy–Krause variation
本文针对d维协变量非参数回归问题，提出了两种最小二乘估计量：完全单调估计量和约束Hardy-Krause变分估计量，它们是单变量保序回归和全变分去噪的多元推广，在平方误差损失下有限样本风险有上界n^{-2/3}（忽略对数因子），从而在一定程度上避免了维数灾难，特别适用于拟合矩形分段常数函数。
We consider the problem of nonparametric regression when the covariate is d dimensional, where d≥1. In this paper, we introduce and study two nonparametric least squares estimators (LSEs) in this setting—the entirely monotonic LSE and the constrained Hardy–Krause variation LSE. We show that these two LSEs are natural generalizations of univariate isotonic regression and univariate total variation denoising, respectively, to multiple dimensions. We discuss the characterization and computation of these two LSEs obtained from n data points. We provide a detailed study of their risk properties under the squared error loss and fixed uniform lattice design. We show that the finite sample risk of these LSEs is always bounded from above by n−2/3 modulo logarithmic factors depending on d; thus these nonparametric LSEs avoid the curse of dimensionality to some extent. We also prove nearly matching minimax lower bounds. Further, we illustrate that these LSEs are particularly useful in fitting rectangular piecewise constant functions. Specifically, we show that the risk of the entirely monotonic LSE is almost parametric (at most 1/n up to logarithmic factors) when the true function is well approximable by a rectangular piecewise constant entirely monotone function with not too many constant pieces. A similar result is also shown to hold for the constrained Hardy–Krause variation LSE for a simple subclass of rectangular piecewise constant functions. We believe that the proposed LSEs yield a novel approach to estimating multivariate functions using convex optimization that avoid the curse of dimensionality to some extent.