SGL and CLOT Algorithms Exhibit Both Near-Ideal Behavior and Grouping Effect

In sparse regression, the LASSO algorithm exhibits near-ideal behavior, but not the grouping effect (assigning nearly equal weights to highly correlated features). In the other direction, the Elastic Net (EN) algorithm exhibits the grouping effect but not near-ideal behavior. In a companion paper by the present authors, it is shown that the Sparse Group Lasso (SGL) algorithm exhibits near-ideal behavior provided the maximum size of the groups is sufficiently small. In this paper it is shown that the SGL algorithm exhibits near-ideal behavior without any restrictions on the group sizes. In addition, it is also shown that the SGL algorithm assigns nearly equal weights to highly correlated features within the same group, but not necessarily to highly correlated features in distinct groups. As a corollary, the algorithm resulting from choosing the entire feature set as a single group, referred to here as CLOT (Combined L-One and Two) also exhibits both near-ideal behavior as well as the grouping effect. The CLOT algorithm is applied to predict the IC50 values of 116 chemical compounds on 72 lung cancer cell lines. The results show that CLOT achieves greater accuracy than EN in 105 out of the 116 regression problems, and greater accuracy than LASSO in every single regression problem. Finally, some results on compressibility in the CLOT norm are derived, which intermediate between two sets of known results, namely compressibility with respect to the `1-norm and incompressibility with respect to the `2-norm.