Towards a New Matrix Decomposition

Low rank matrix approximations have many applications in different domains. In system theory it has been used in model reduction schemes, in system identification with outputerror models and in static errors-in-variables problems, for instance. The approximations are mostly performed using the singular value decomposition. This is optimal for all unitarily invariant matrix norms, such as the Frobenius norm. From a statistical point of view it is justified when the components are perturbed by independent and identically distributed zero mean Gaussian noise. If this assumption is not valid other norms and thus approximations should be considered. Below we consider the l1-norm that is optimal if the noise samples follow the Laplace or double-exponential distribution and we indicate how to obtain for an arbitrary matrix, the optimal decomposition -similar to the singular value decompositionassociated with this norm.