# Convex Modeling of Interactions with Strong Heredity

arXiv: https://arxiv.org/abs/1410.3517 · HTML: https://mlsanity.com/paper/1410.3517/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Oct 2014.

## Built on by (most cited first, 6 of 6)

- [The SKIM-FA Kernel: High-Dimensional Variable Selection and Nonlinear Interaction Discovery in Linear Time](https://mlsanity.com/paper/2106.12408/) (2021)
- [Hierarchical shrinkage Gaussian processes: applications to computer code emulation and dynamical system recovery](https://mlsanity.com/paper/2302.00755/) (2023)
- [An ADMM approach for multi-response regression with overlapping groups and interaction effects](https://mlsanity.com/paper/2303.11155/) (2023)
- [Structured Learning in Time-dependent Cox Models](https://mlsanity.com/paper/2306.12528/) (2023)
- [Optimal Kernel Learning for Gaussian Process Models with High-Dimensional Input](https://mlsanity.com/paper/2502.16617/) (2025)
- [CUBE: Contrastive Understanding by Balanced Experiments](https://mlsanity.com/paper/2509.10825/) (2025)

## Abstract (arXiv)

We consider the task of fitting a regression model involving interactions among a potentially large set of covariates, in which we wish to enforce strong heredity. We propose FAMILY, a very general framework for this task. Our proposal is a generalization of several existing methods, such as VANISH [Radchenko and James, 2010], hierNet [Bien et al., 2013], the all-pairs lasso, and the lasso using only main effects. It can be formulated as the solution to a convex optimization problem, which we solve using an efficient alternating directions method of multipliers (ADMM) algorithm. This algorithm has guaranteed convergence to the global optimum, can be easily specialized to any convex penalty function of interest, and allows for a straightforward extension to the setting of generalized linear models. We derive an unbiased estimator of the degrees of freedom of FAMILY, and explore its performance in a simulation study and on an HIV sequence data set.