Relaxed regularization for linear inverse problems

Abstract

We consider regularized least-squares problems of the form min x 1/2 kAx − bk22+ R(Lx). Recently, Zheng et al. [45] proposed an algorithm called Sparse Relaxed Regularized Regression (SR3) that employs a splitting strategy by introducing an auxiliary variable y and solves minx,y 1 2 kAx − bk22 + k/2 2 kLx − yk22 + R(x). By minimizing out the variable x, we obtain an equivalent optimization problem miny 1 2 kFy − gk22 + R(y). In our work, we view the SR3 method as a way to approximately solve the regularized problem. We analyze the conditioning of the relaxed problem in general and give an expression for the SVD of F as a function of κ. Furthermore, we relate the Pareto curve of the original problem to the relaxed proble