  Lasso is a seminal contribution to high-dimensional statistics, but it hinges
on a tuning parameter that is difficult to calibrate in practice. A partial
remedy for this problem is Square-Root Lasso, because it inherently calibrates
to the noise variance. However, Square-Root Lasso still requires the
calibration of a tuning parameter to all other aspects of the model. In this
study, we introduce TREX, an alternative to Lasso with an inherent calibration
to all aspects of the model. This adaptation to the entire model renders TREX
an estimator that does not require any calibration of tuning parameters. We
show that TREX can outperform cross-validated Lasso in terms of variable
selection and computational efficiency. We also introduce a bootstrapped
version of TREX that can further improve variable selection. We illustrate the
promising performance of TREX both on synthetic data and on a recent
high-dimensional biological data set that considers riboflavin production in B.
subtilis.
