Relaxed regularization for linear inverse problems

Abstract

We consider regularized least-squares problems of the form ${\min }_x\frac{1}{2}\Vert Ax-b{\Vert }_2^2+\mathcal{R}(Lx)$. Recently, Zheng et al., 2019, proposed an algorithm called Sparse Relaxed Regularized Regression (SR3) that employs a splitting strategy by introducing an auxiliary variable $y$ and solves ${\min }_{x,y}\frac{1}{2}\Vert Ax-b{\Vert }_2^2+\frac{\kappa }{2}\Vert Lx-y{\Vert }_2^2+\mathcal{R}(x)$. By minimizing out the variable $x$ we obtain an equivalent system ${\min }_y\frac{1}{2}\Vert F_{\kappa }y-g_{\kappa }{\Vert }_2^2+\mathcal{R}(y)$. 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 $F_{\kappa }$ as a function of $\kappa$. Furthermore, we relate the Pareto curve of the original problem to the relaxed problem and we quantify the error incurred by relaxation in terms of $\kappa$. Finally, we propose an efficient iterative method for solving the relaxed problem with inexact inner iterations. Numerical examples illustrate the approach.

25 pages, 14 figures, submitted to SIAM Journal for Scientific Computing special issue Sixteenth Copper Mountain Conference on Iterative Methods