Optimal transform formed by a combination of nonlinear operators: the case of data dimensionality reduction
Anatoli Torokhti, Phil George Howlett · IEEE Transactions on Signal Processing · 2006
In this paper, a new approach to constructing optimal nonlinear transforms of random vectors is proposed and justified. The proposed transform T/sub 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 T/sub p/ are represented as a combination of three operations F/sub k/, Q/sub k/, and /spl psi//sub k/ with k=1,...,p. The prime idea is to determine F/sub k/ separately, for each k=1,...,p, from an associated rank-constrained minimization problem similar to that used in the Karhunen-Loe/spl grave/ve transform (KLT). The operations Q/sub k/ and /spl psi//sub k/ are auxiliary for finding F/sub k/. The contribution of each term in T/sub p/ improves the entire transform performance. A rigorous analysis of errors associated with the proposed transforms is given. It is shown that the proposed transform improves such characteristics of rank-reduced transforms as compression ratio and the accuracy of decompression and reduces required computational work. The Fourier series in Hilbert space, the Wiener filter, the KLT and the transforms presented in earlier author papers in the field are particular cases of the proposed method. Theoretical results are illustrated with numerical simulations.