Combined Reduced-Rank Transform

Abstract:
Our work focuses on study of new mathematical models (also called transforms) for nonlinear phenomena arising in real world problems. A transform is treated
as a mathematical representation of a phenomenon. The mathematical treatment is naturally associated with the concept of operators. In particular, a phenomenon
can be described by a special linear operator with non-complete (reduced) rank.

We propose and justify a new approach to constructing optimal nonlinear transforms of random vectors. It is shown that the proposed  transform improves such characteristics of the known rank-reduced transforms  as compression ratio, accuracy of decompression and required computational work.A device used in achieving such improvements is based on the presentation of the proposed transform  in the form of the sum where each term is interpreted as a particular rank-reduced transform. Moreover, each term is represented as a combination of three operations.The prime idea is to determine specific operators, which comprise each term, separately from an associated rank-constrained minimization problem similar to that in the Karhunen-Loeve transform.

The corresponding unconstrained nonlinear optimal transform is also considered. Such a transform is important in its own right because it is treated as an optimal filter without information compression. The Wiener filter is a particular case of the presented unconstrained transform.

A rigorous analysis of errors associated with the proposed transforms is given.