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 $\varphi_k$ with $k=1,\dots,p$. The prime idea is to determine $\mathcal F_k$ separately, for each $k=1,\dots,p$, from an associated rank-constrained minimization problem similar to that used in the Karhunen–Loève transform. The operations $\mathcal Q_k$ and $\varphi_k$ 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.