# Model Selection for High Dimensional Quadratic Regression via Regularization

arXiv: https://arxiv.org/abs/1501.00049 · HTML: https://mlsanity.com/paper/1501.00049/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Dec 2014.

## Built on by (most cited first, 8 of 8)

- [Discovering Categorical Main and Interaction Effects Based on Association Rule Mining](https://mlsanity.com/paper/2104.04728/) (2021)
- [Random Intersection Chains](https://mlsanity.com/paper/2104.04714/) (2021)
- [Multi-task Learning for Gaussian Graphical Regressions with High Dimensional Covariates](https://mlsanity.com/paper/2205.10672/) (2022)
- [Exploring hyperelastic material model discovery for human brain cortex: multivariate analysis vs. artificial neural network approaches](https://mlsanity.com/paper/2310.10762/) (2023)
- [Pre-processing with Orthogonal Decompositions for High-dimensional Explanatory Variables](https://mlsanity.com/paper/2106.09071/) (2021)
- [Online Interaction Detection for Click-Through Rate Prediction](https://mlsanity.com/paper/2106.15400/) (2021)
- [Sparsified Simultaneous Confidence Intervals for High-Dimensional Linear Models](https://mlsanity.com/paper/2307.07574/) (2023)
- [Sparse Bayesian Modeling of EEG Channel Interactions Improves P300 Brain-Computer Interface Performance](https://mlsanity.com/paper/2602.17772/) (2026)

## Abstract (arXiv)

Quadratic regression (QR) models naturally extend linear models by considering interaction effects between the covariates. To conduct model selection in QR, it is important to maintain the hierarchical model structure between main effects and interaction effects. Existing regularization methods generally achieve this goal by solving complex optimization problems, which usually demands high computational cost and hence are not feasible for high dimensional data. This paper focuses on scalable regularization methods for model selection in high dimensional QR. We first consider two-stage regularization methods and establish theoretical properties of the two-stage LASSO. Then, a new regularization method, called Regularization Algorithm under Marginality Principle (RAMP), is proposed to compute a hierarchy-preserving regularization solution path efficiently. Both methods are further extended to solve generalized QR models. Numerical results are also shown to demonstrate performance of the methods.