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.