The Trimmed Lasso: Sparsity and Robustness

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. 2. Further, in relating the trimmed Lasso to commonly used sparsity-inducing penalty functions, we provide a succinct characterization of the connection between trimmed-Lassolike approaches and penalty functions that are coordinate-wise separable, showing that the trimmed penalties subsume existing coordinate-wise separable penalties, with strict containment in general. 3. Finally, we describe a variety of exact and heuristic algorithms, both existing and new, for trimmed Lasso regularized estimation problems. We include a comparison between the different approaches and an accompanying implementation of the algorithms.