On the Semisupervised Joint Trained Elastic Net

The elastic net (supervised enet henceforth) is a popular and computationally efficient approach for performing the simultaneous tasks of selecting variables, decorrelation, and shrinking the coefficient vector in the linear regression setting. Semisupervised regression, currently unrelated to the supervised enet, uses data with missing response values (unlabeled) along with labeled data to train the estimator. In this article, we propose the joint trained elastic net (jt-enet), which elegantly incorporates the benefits of semisupervised regression with the supervised enet. The supervised enet and other approaches like it rely on shrinking the linear estimator in a way that simultaneously performs variable selection and decorrelates the data. Both the variable selection and decorrelation components of the supervised enet inherently rely on the pairwise correlation structure in the feature data. In circumstances in which the number of variables is high, the feature data are relatively easy to obtain, and the response is expensive to generate, it seems reasonable that one would want to be able to use any existing unlabeled observations to more accurately define these correlations. However, the supervised enet is not able to incorporate this information and focuses only on the information within the labeled data. In this article, we propose the jt-enet, which allows the unlabeled data to influence the variable selection, decorrelation, and shrinkage capabilities of the linear estimator. In addition, we investigate the impact of unlabeled data on the risk and bias of the proposed estimator. The jt-enet is demonstrated on two applications with encouraging results. Online supplementary material is available for this article.