Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 039, 21 pages      math.OC/0604220

Combined Reduced-Rank Transform

Anatoli Torokhti and Phil Howlett
School of Mathematics and Statistics, University of South Australia, Australia

Received November 25, 2005, in final form March 22, 2006; Published online April 07, 2006

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 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 φk 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-Loève transform. The operations Qk andφk are auxiliary for finding Fk. The contribution of each term in Tp 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.

Key words: best approximation; Fourier series in Hilbert space; matrix computation.

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