MARS via LASSO

TL;DR

This paper introduces a LASSO-based variant of the MARS method for nonparametric regression, which achieves favorable convergence rates and is computationally feasible through convex optimization, improving over traditional MARS.

Contribution

The paper proposes a novel LASSO-inspired MARS variant that leverages convex optimization and offers theoretical convergence guarantees under standard assumptions.

Findings

Achieves near-logarithmic dependence on dimension in convergence rate

Connects MARS variant to smoothness-based nonparametric estimation

Demonstrates improved performance over traditional MARS in simulations and real data

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.