Combined Reduced-Rank Transform

We propose and justify a new approach to constructing optimal nonlinear transforms of random vectors. We show that the proposed transform improves such characteristics of rank-reduced transforms as compression ratio, accuracy of decompression and reduces required computational work. The proposed transform ${\mathcal T}_p$ is presented in the form of a sum with $p$ terms where each term is interpreted as a particular rank-reduced transform. Moreover, terms in ${\mathcal T}_p$ are represented as a combination of three operations ${\mathcal F}_k$, ${\mathcal Q}_k$ and ${\boldsymbol{\phi}}_k$ with $k=1,...,p$. The prime idea is to determine ${\mathcal F}_k$ separately, for each $k=1,...,p$, from an associated rank-constrained minimization problem similar to that used in the Karhunen--Lo\`{e}ve transform. The operations ${\mathcal Q}_k$ and ${\boldsymbol{\phi}}_k$ are auxiliary for finding ${\mathcal F}_k$. The contribution of each term in ${\mathcal T}_p$ improves the entire transform performance. A 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 signal compression. A rigorous analysis of errors associated with the proposed transforms is given.