Quantile universal threshold: model selection at the detection edge for high-dimensional linear regression

TL;DR

This paper introduces Quantile Universal Thresholding, a method for selecting the regularization parameter in high-dimensional linear regression that balances true positive and false discovery rates effectively.

Contribution

The paper proposes a novel thresholding method for model selection at the detection edge, improving variable selection and predictive performance in high-dimensional settings.

Findings

Achieves high true positive rate and low false discovery rate.

Demonstrates effectiveness through extensive simulations and real data.

Provides a practical approach for threshold selection in sparse models.

Abstract

To estimate a sparse linear model from data with Gaussian noise, consilience from lasso and compressed sensing literatures is that thresholding estimators like lasso and the Dantzig selector have the ability in some situations to identify with high probability part of the significant covariates asymptotically, and are numerically tractable thanks to convexity. Yet, the selection of a threshold parameter $\lambda$ remains crucial in practice. To that aim we propose Quantile Universal Thresholding, a selection of $\lambda$ at the detection edge. We show with extensive simulations and real data that an excellent compromise between high true positive rate and low false discovery rate is achieved, leading also to good predictive risk.