Using structured low-rank approximation for sparse signal recovery

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 the low-rank prior. This paper considers a modified 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 structured low-rank approximation with missing data and propose a solution methods based on the variable projections principle. We compare the structured low-rank approximation approach with the classical sparsity inducing method of 1-norm regularization. The 1-norm regularization method is effective for sum-of-exponentials modeling with a large number of samples, however, it is not suitable for identification of systems with damping.


  1. I. Markovsky and P.-L. Dragotti. Using structured low-rank approximation for sparse signal recovery. In Latent Variable Analysis and Signal Separation, Lecture Notes in Computer Science, pages 479–487. Springer, 2018.
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