A Generalized Elastic Net Regularization with Smoothed <i>l</i><sub>0</sub> Penalty

Abstract

This paper presents an accurate and efficient algorithm for solving the generalized elastic net regularization problem with smoothed 0  penalty for recovering sparse vector. Finding the optimal solution to the unconstrained 0  minimization problem in the recovery of compressive sensed signals is an NP-hard problem. We proposed an iterative algorithm to solve this problem. We then prove that the algorithm is convergent based on algebraic methods. The numerical result shows the efficiency and the accuracy of the algorithm.

The use of grossone in elastic net regularization and sparse support vector machines

“…In [18], in the context of elastic net regularization (discussed in detail in Section 4), the authors proposed the following approximation:…”

“…where λ 0 > 0 and λ 2 > 0 are regularization parameters. In [18] the authors suggest to substitute x 0 with x 0,δ , as defined in (7), for fixed positive δ, and a convergent algorithm for the solution of the corresponding optimization problem is proposed. Clearly the obtained solution only approximates the optimal solution of (10).…”

“…We propose to use in (10) the approximation (8) of x 0 . Replicating some of the proofs in [18] a convergent algorithm can be constructed. Moreover, due to the position (9), apart from terms of order x −1 or below, the obtained solution solves also problem (10).…”