Isotonic regression in general dimensions

By Qiyang Han, Tengyao Wang, Sabyasachi Chatterjee, Richard J. Samworth. Annals of Statistics, 2017

111 citations recorded by the source corpus.

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 poly-logarithmic 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 poly-logarithmic factors. Previous results are confined to the case $d \leq 2$. 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 poly-logarithmic 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.

Subfields: Artificial Intelligence, Computational Mechanics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/60c42451891e56d12127e9e8535fd3c3a97ca3e4

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Citations in the corpus

Nonparametric Shape-Restricted Regression (2017, background): “[69], studied d-dimensional isotonic regression for general d ≥ 1 and proved versions of inequalities (27) and (29).”