Using hankel structured low-rank approximation for sparse signal recovery

Abstract

Structured low-rank approximation is used in model reduction, system identification, and signal processing to find low-complexity models from data. The rank constraint imposes the condition that the approximation has bounded complexity and the optimization criterion aims to find the best match between the data—a trajectory of the system—and the approximation. In some applications, however, the data is sub-sampled from a trajectory, which poses the problem of sparse approximation using the low-rank prior. This paper considers a modified Hankel structured low-rank approximation problem where the observed data is a linear transformation of a system’s trajectory with reduced dimension. We reformulate this problem as a Hankel structured low-rank approximation with missing data and propose a solution methods based on the variable projections principle. We compare the Hankel structured low-rank approximation approach with the classical sparsity inducing method of l1 -norm regularization. The l1 -norm regularization method is effective for sum-of-exponentials modeling with a large number of samples, however, it is not suitable for damped system identification.