Structured low-rank approximation and its applications

Fitting data by a bounded complexity linear model is equivalent to low-rank approximation of a matrix constructed from the data. The data matrix being Hankel structured is equivalent to the existence of a linear time-invariant system that fits the data and the rank constraint is related to a bound on the model complexity. In the special case of fitting by a static model, the data matrix and its low-rank approximation are unstructured. We outline applications in system theory (approximate realization, model reduction, output error and errors-in-variables identification), signal processing (harmonic retrieval, sum-of-damped exponentials and finite impulse response modeling), and computer algebra (approximate common divisor). Algorithms based on the variable projections and alternating projections methods are presented. Generalizations of the low-rank approximation problem result from different approximation criteria (e.g., weighted norm), constraints on the data matrix (e.g., nonnegativity), and data structures (e.g., kernel mapping). Related problems are rank minimization and structured pseudospectra.