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Nonparametric regression with adaptive truncation via a convex hierarchical penalty

Author

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  • Asad Haris
  • Ali Shojaie
  • Noah Simon

Abstract

SUMMARY We consider the problem of nonparametric regression with a potentially large number of covariates. We propose a convex, penalized estimation framework that is particularly well suited to high-dimensional sparse additive models and combines the appealing features of finite basis representation and smoothing penalties. In the case of additive models, a finite basis representation provides a parsimonious representation for fitted functions but is not adaptive when component functions possess different levels of complexity. In contrast, a smoothing spline-type penalty on the component functions is adaptive but does not provide a parsimonious representation. Our proposal simultaneously achieves parsimony and adaptivity in a computationally efficient way. We demonstrate these properties through empirical studies and show that our estimator converges at the minimax rate for functions within a hierarchical class. We further establish minimax rates for a large class of sparse additive models. We also develop an efficient algorithm that scales similarly to the lasso with the number of covariates and sample size.

Suggested Citation

  • Asad Haris & Ali Shojaie & Noah Simon, 2019. "Nonparametric regression with adaptive truncation via a convex hierarchical penalty," Biometrika, Biometrika Trust, vol. 106(1), pages 87-107.
  • Handle: RePEc:oup:biomet:v:106:y:2019:i:1:p:87-107.
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    File URL: http://hdl.handle.net/10.1093/biomet/asy056
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    Cited by:

    1. Bhatnagar, Sahir R. & Lu, Tianyuan & Lovato, Amanda & Olds, David L. & Kobor, Michael S. & Meaney, Michael J. & O'Donnell, Kieran & Yang, Archer Y. & Greenwood, Celia M.T., 2023. "A sparse additive model for high-dimensional interactions with an exposure variable," Computational Statistics & Data Analysis, Elsevier, vol. 179(C).

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