# High-dimensional variable selection via low-dimensional adaptive learning

## TL;DR

The paper introduces AdaSub, a stochastic search algorithm for high-dimensional variable selection that adaptively solves low-dimensional sub-problems, effectively identifying optimal models based on criteria like BIC in complex datasets.

## Contribution

It proposes a novel adaptive subspace method for variable selection in high-dimensional linear regression, with proven convergence properties and demonstrated superior performance.

## Key findings

- AdaSub converges to the best model under certain conditions.
- The method outperforms alternative variable selection techniques in simulations.
- Effective in high-dimensional real data applications.

## Abstract

A stochastic search method, the so-called Adaptive Subspace (AdaSub) method, is proposed for variable selection in high-dimensional linear regression models. The method aims at finding the best model with respect to a certain model selection criterion and is based on the idea of adaptively solving low-dimensional sub-problems in order to provide a solution to the original high-dimensional problem. Any of the usual $\ell_0$-type model selection criteria can be used, such as Akaike's Information Criterion (AIC), the Bayesian Information Criterion (BIC) or the Extended BIC (EBIC), with the last being particularly suitable for high-dimensional cases. The limiting properties of the new algorithm are analysed and it is shown that, under certain conditions, AdaSub converges to the best model according to the considered criterion. In a simulation study, the performance of AdaSub is investigated in comparison to alternative methods. The effectiveness of the proposed method is illustrated via various simulated datasets and a high-dimensional real data example.