## The Annals of Statistics

### Minimax-optimal nonparametric regression in high dimensions

#### Abstract

Minimax $L_{2}$ risks for high-dimensional nonparametric regression are derived under two sparsity assumptions: (1) the true regression surface is a sparse function that depends only on $d=O(\log n)$ important predictors among a list of $p$ predictors, with $\log p=o(n)$; (2) the true regression surface depends on $O(n)$ predictors but is an additive function where each additive component is sparse but may contain two or more interacting predictors and may have a smoothness level different from other components. For either modeling assumption, a practicable extension of the widely used Bayesian Gaussian process regression method is shown to adaptively attain the optimal minimax rate (up to $\log n$ terms) asymptotically as both $n,p\to\infty$ with $\log p=o(n)$.

#### Article information

Source
Ann. Statist., Volume 43, Number 2 (2015), 652-674.

Dates
First available in Project Euclid: 3 March 2015

https://projecteuclid.org/euclid.aos/1425398504

Digital Object Identifier
doi:10.1214/14-AOS1289

Mathematical Reviews number (MathSciNet)
MR3319139

Zentralblatt MATH identifier
1312.62052

Subjects
Primary: 62G08: Nonparametric regression 62C20: Minimax procedures
Secondary: 60G15: Gaussian processes

#### Citation

Yang, Yun; Tokdar, Surya T. Minimax-optimal nonparametric regression in high dimensions. Ann. Statist. 43 (2015), no. 2, 652--674. doi:10.1214/14-AOS1289. https://projecteuclid.org/euclid.aos/1425398504

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