Low-Rank Approximation and Its Applications
I. Markovsky

Low-rank approximation is a unifying theme in data modelling. A matrix constructed from the data being rank deficient implies that there is an exact low complexity linear model for that data. Moreover, the rank of the data matrix corresponds to the complexity of the model. In the generic case when an exact low-complexity model does not exist, the aim is to find an model that fits the data approximately. The approach that we present is to find small (in some specified sense) modification of the data that renders the modified data exact. The exact model for the modified data is an optimal (in the specified sense) approximate model for the original data. The corresponding computational problem is low-rank approximation. In the numerical linear algebra literature this approach is referred to as "total least squares" and in the statistical literature as "errors-invariables modelling".

References:

- I. Markovsky, Structured low-rank approximation and its applications, Automatica, 44:891--909, 2007
- I. Markovsky and S. Van Huffel, Overview of total least squares methods, Signal Processing, 87:2283--2302, 2007.
- I. Markovsky, et al., Exact and Approximate Modeling of Linear Systems: A Behavioral Approach, SIAM, 2006.