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October 2019 Isotonic regression in general dimensions
Qiyang Han, Tengyao Wang, Sabyasachi Chatterjee, Richard J. Samworth
Ann. Statist. 47(5): 2440-2471 (October 2019). DOI: 10.1214/18-AOS1753


We study the least squares regression function estimator over the class of real-valued functions on $[0,1]^{d}$ that are increasing in each coordinate. For uniformly bounded signals and with a fixed, cubic lattice design, we establish that the estimator achieves the minimax rate of order $n^{-\min\{2/(d+2),1/d\}}$ in the empirical $L_{2}$ loss, up to polylogarithmic factors. Further, we prove a sharp oracle inequality, which reveals in particular that when the true regression function is piecewise constant on $k$ hyperrectangles, the least squares estimator enjoys a faster, adaptive rate of convergence of $(k/n)^{\min(1,2/d)}$, again up to polylogarithmic factors. Previous results are confined to the case $d\leq2$. Finally, we establish corresponding bounds (which are new even in the case $d=2$) in the more challenging random design setting. There are two surprising features of these results: first, they demonstrate that it is possible for a global empirical risk minimisation procedure to be rate optimal up to polylogarithmic factors even when the corresponding entropy integral for the function class diverges rapidly; second, they indicate that the adaptation rate for shape-constrained estimators can be strictly worse than the parametric rate.


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Qiyang Han. Tengyao Wang. Sabyasachi Chatterjee. Richard J. Samworth. "Isotonic regression in general dimensions." Ann. Statist. 47 (5) 2440 - 2471, October 2019.


Received: 1 August 2017; Revised: 1 April 2018; Published: October 2019
First available in Project Euclid: 3 August 2019

zbMATH: 07114918
MathSciNet: MR3988762
Digital Object Identifier: 10.1214/18-AOS1753

Primary: 62C20 , 62G05 , 62G08

Keywords: Adaptation , block increasing functions , isotonic regression , least squares , Sharp oracle inequality , statistical dimension

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.47 • No. 5 • October 2019
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