Best approximation of the identity mapping: The case of variable finite memory
Journal of Approximation Theory
Towards theory of generic Principal Component Analysis
Journal of Multivariate Analysis
| Hi-index | 35.68 |
In this paper, a new approach to constructing optimal nonlinear transforms of random vectors is proposed and justified. The proposed transform Tp 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 Tp are represented as a combination of three operations Fk, Qk, and ψk with k=1,...,p. The prime idea is to determine Fk separately, for each k=1,...,p, from an associated rank-constrained minimization problem similar to that used in the Karhunen-Loe`ve transform (KLT). The operations Qk and ψk are auxiliary for finding Fk. The contribution of each term in Tp 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.