Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy–Krause variation
通过完全单调性与Hardy–Krause变差推广多元保序回归和总变差去噪AI 解读 ↓
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通过完全单调性与Hardy–Krause变差推广多元保序回归和总变差去噪
- 研究问题
- 如何将一元保序回归和总变差去噪推广到多维非参数回归，并在形状约束下控制风险、适应矩形分段常数结构？
- 方法与思路
- 作者研究完全单调与受约束Hardy–Krause变差最小二乘估计，讨论计算，并在固定均匀格点设计下分析平方误差风险。
- 材料陈述的贡献
- 摘要称得到有限样本风险上界与近乎匹配的极小极大下界，并说明部分矩形分段常数函数下可获得近参数级风险表现。
- 阅读关注点
- 应核对两种函数约束、设计与风险定义、维度相关对数因子、分段常数适应条件及凸优化实现；风险速率须查原文。
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原文摘要
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.