# MARS via LASSO

arXiv: https://arxiv.org/abs/2111.11694 · HTML: https://mlsanity.com/paper/2111.11694/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 29 references that are in the map.
- First posted on arXiv 23 Nov 2021.

## Builds on (most cited first, 7 of 7)

- [Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks](https://mlsanity.com/paper/2109.08844/) (2021)
- [Exact Support Recovery for Sparse Spikes Deconvolution](https://mlsanity.com/paper/1306.6909/) (2013)
- [The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy](https://mlsanity.com/paper/1811.06416/) (2018)
- [Efficient Estimation of Pathwise Differentiable Target Parameters with the Undersmoothed Highly Adaptive Lasso](https://mlsanity.com/paper/1908.05607/) (2019)
- [Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy-Krause variation](https://mlsanity.com/paper/1903.01395/) (2019)
- [Fused Lasso Additive Model](https://mlsanity.com/paper/1409.5391/) (2014)
- [Hierarchical Total Variations and Doubly Penalized ANOVA Modeling for Multivariate Nonparametric Regression](https://mlsanity.com/paper/1906.06729/) (2019)

## Built on by (most cited first, 5 of 5)

- [Adaptive debiased machine learning using data-driven model selection techniques](https://mlsanity.com/paper/2307.12544/) (2023)
- [Multivariate Trend Filtering for Lattice Data](https://mlsanity.com/paper/2112.14758/) (2021)
- [A Cross Validation Framework for Signal Denoising with Applications to Trend Filtering, Dyadic CART and Beyond](https://mlsanity.com/paper/2201.02654/) (2022)
- [Spatially Adaptive Online Prediction of Piecewise Regular Functions](https://mlsanity.com/paper/2203.16587/) (2022)
- [Adaptive Bayesian Online Learning via Expert Aggregation](https://mlsanity.com/paper/2607.20239/) (2026)

## Abstract (arXiv)

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.