Linear regression via elastic net: non-enumerative leave-one-out verification of feature selection

The feature-selective non-quadratic Elastic Net criterion of regression estimation is completely determined by two numerical regularization parameters which penalize, respectively, the squared and absolute values of the regression coefficients under estimation. It is an inherent property of the minimum of the Elastic Net that the values of regularization parameters completely determine a partition of the variable set into three subsets of negative, positive, and strictly zero values, so that the former two subsets and the latter subset are, respectively, associated with “informative” and “redundant” features. We propose in this paper to treat this partition as a secondary structural parameter to be verified by leave-one-out cross validation. Once the partitioning is fixed, we show that there exists a non-enumerative method for computing the leave-one-out error rate, thus enabling an evaluation of model generality in order to tune the structural parameters without the necessity of multiple training repetitions.