Tensor approximation and signal processing applications

Abstract

Recent advances in application fields such as blind deconvolution and independent component analysis have revived the interest in higher-order statistics (HOS), showing them to be an indispensable tool for providing piercing solutions to these problems. Multilinear algebra and in particular tensor analysis provide a natural mathematical framework for studying HOS in view of the inherent multiindexing structure and the multilinearity of cumulants of linear random processes. The aim of this paper is twofold. First, to provide a semitutorial exposition of recent developments in tensor analysis and blind HOS-based signal processing. Second, to expose the strong links connecting recent results on HOS blind deconvolution with parallel, seemingly unrelated, advances in tensor approximation. In this context, we provide a detailed convergence analysis of a recently introduced tensor-equivalent of the power method for determining a rank-1 approximant to a matrix and we develop a novel version adapted to the symmetric case. A new effective initialization is also proposed which has been shown to frequently outperform that induced by the Tensor Singular-Value Decomposition (TSVD) in terms of its closeness to the globally optimal solution. In light of the equivalence of the symmetric high-order power method with the so-called superexponential algorithm for blind deconvolution, also revealed here, the effectiveness of the novel initialization shows great promise in yielding a clear solution to the local-extrema problem, ubiquitous in HOS-based blind deconvolution approaches. E. Kofidis was supported by a Training and Mobility of Researchers (TMR) grant of the European Commission, contract no. ERBFMBICT982959. c ©0000 (copyright holder)