MARS via LASSO
Dohyeong Ki, Billy Fang, Adityanand Guntuboyina
The Annals of Statistics · 2024 · Volume 52 · Issue 3
Abstract
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 cross-validation scheme for the selection of the involved tuning parameter and compare it to the usual MARS method in various simulation and real data settings.
Keywords
62G08, Bracketing entropy bounds, constrained least squares estimation, curse of dimensionality, Hardy–Krause variation, infinite-dimensional optimization, integrated Brownian sheet, L1 penalty, locally adaptive regression spline, metric entropy bounds, mixed derivatives, Nonparametric regression, piecewise linear function estimation, Small ball probability, Tensor products, Total variation regularization, Trend filtering
Enable JavaScript for citation tools and additional paper interactions.