Recent progress on variable projection methods for structured low-rank approximation

Rank deficiency of a data matrix is equivalent to the existence of an exact linear model for the data. For the purpose of linear static modeling, the matrix is unstructured and the corresponding modeling problem is an approximation of the matrix by another matrix of a lower rank. In the context of linear time-invariant dynamic models, the appropriate data matrix is Hankel and the corresponding modeling problems becomes structured low-rank approximation. Low-rank approximation has applications in: system identification; signal processing, machine learning, and computer algebra, where different types of structure and constraints occur. This paper gives an overview of recent progress in efficient local optimization algorithms for solving weighted mosaic-Hankel structured low-rank approximation problems. In addition, the data matrix may have missing elements and elements may be specified as exact. The described algorithms are implemented in a publicly available software package. Their application to system identification, approximate common divisor, and data-driven simulation problems is described in this paper and is illustrated by reproducible simulation examples. As a data modeling paradigm the low-rank approximation setting is closely related to the behavioral approach in systems and control, total least squares, errors-in-variables modeling, principal component analysis, and rank minimization.