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.

翻译：多元自适应回归样条（MARS）是Friedman于1991年提出的非参数回归经典方法，通过拟合简单的非线性与非可加函数来处理回归数据。本研究提出并系统研究了一种基于LASSO的自然变体MARS方法。该方法在凸函数类上实施最小二乘估计，通过考虑MARS基函数构成的无限维线性组合并施加基于变分复杂度的约束条件实现。尽管被定义为无限维优化问题的解，所提估计量可通过有限维凸优化计算获得。在若干标准设计假设下，我们证明该估计量实现的对数维数收敛速率，能在一定程度上规避传统的维数灾难问题。同时证明该方法与基于光滑性约束的非参数估计技术存在天然关联。我们采用交叉验证策略实现调节参数的选择，并通过模拟实验与真实数据对比将所提方法与经典MARS方法进行系统比较。