The Trimmed Lasso: Sparsity and Robustness

Understand

Nonconvex penalty methods for sparse modeling in linear regression have been a topic of fervent interest in recent years.

• Herein, we study a family of nonconvex penalty functions that we call the trimmed Lasso and that offers exact control over the desired level of sparsity of estimators.
• We analyze its structural properties and in doing so show the following: 1) Drawing parallels between robust statistics and robust optimization, we show that the trimmed-Lasso-regularized least squares problem can be viewed as a generalized form of total least squares under a specific model of uncertainty.
• In contrast, this same model of uncertainty, viewed instead through a robust optimization lens, leads to the convex SLOPE (or OWL) penalty.