Approximate regularization paths for nuclear norm minimization using singular value bounds

The widely used nuclear norm heuristic for rank minimization problems introduces a regularization parameter which is difficult to tune. We have recently proposed a method to approximate the regularization path, i.e., the optimal solution as a function of the parameter, which requires solving the problem only for a sparse set of points. In this paper, we extend the algorithm to provide error bounds for the singular values of the approximation. We exemplify the algorithms on large scale benchmark examples in model order reduction. Here, the order of a dynamical system is reduced by means of constrained minimization of the nuclear norm of a Hankel matrix.