MARS VIA LASSO

Dohyeong Ki

doi:10.1214/24-AOS2384

2024-06-01

Dohyeong Ki *

Billy Fang

Adityanand Guntuboyina

DOI：10.1214/24-AOS2384

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摘要

En 中文

Multivariate adaptive regression splines (MARS) is a popular method for nonparametric regression introduced by Friedman in 1991. MARS fits simple nonlinear and non-additive functions to regression data. We propose and study a natural lasso variant of the MARS method. Our method is based on least squares estimation over a convex class of functions obtained by considering infinite-dimensional linear combinations of functions in the MARS basis and imposing a variation based complexity constraint. Our estimator can be computed via finite-dimensional convex optimization, although it is defined as a solution to an infinite-dimensional optimization problem. Under a few standard design assumptions, we prove that our estimator achieves a rate of convergence that depends only logarithmically on dimension and thus avoids the usual curse of dimensionality to some extent. We also show that our method is naturally connected to nonparametric estimation techniques based on smoothness constraints. We implement our method with a crossvalidation scheme for the selection of the involved tuning parameter and compare it to the usual MARS method in various simulation and real data settings.

Keyword:

Bracketing entropy bounds

constrained least squares estimation

curse of dimension- ality

Hardy-Krause variation

infinite-dimensional optimization

integrated Brownian sheet

locally adaptive re- gression spline

L1 penalty

metric entropy bounds

mixed derivatives

nonparametric regression