On the best rank-1 approximation of higher-order supersymmetric tensors

Abstract. Recently the problem of determining the best, in the least-squares sense, rank-1 approximation to a higher-order tensor was studied and an iterative method that extends the wellknown power method for matriceswasproposed for itssolution. Thishigher-order power method is also proposed for the special but important class of supersymmetric tensors, with no change. A simplified version, adapted to the special structure of the supersymmetric problem, is deemed unreliable, asitsconvergence isnot guaranteed. The aim of thispaper isto show that a symmetric version of the above method converges under assumptions of convexity (or concavity) for the functional induced by the tensor in question, assumptions that are very often satisfied in practical applications. The use of this version entails significant savings in computational complexity as compared to the unconstrained higher-order power method. Furthermore, a novel method for initializing the iterative processisdeveloped which hasbeen observed to yield an estimate that liescloser to the global optimum than the initialization suggested before. Moreover, its proximity to the global optimum is a priori quantifiable. In the course of the analysis, some important properties that the supersymmetry of a tensor implies for its square matrix unfolding are also studied

Blind Source Separation: Fundamentals and Recent Advances (A Tutorial Overview Presented at SBrT-2001)

Blind source separation (BSS), i.e., the decoupling of unknown signals that have been mixed in an unknown way, has been a topic of great interest in the signal processing community for the last decade, covering a wide range of applications in such diverse fields as digital communications, pattern recognition, biomedical engineering, and financial data analysis, among others. This course aims at an introduction to the BSS problem via an exposition of well-known and established as well as some more recent approaches to its solution. A unified way is followed in presenting the various results so as to more easily bring out their similarities/differences and emphasize their relative advantages/disadvantages. Only a representative sample of the existing knowledge on BSS will be included in this course. The interested readers are encouraged to consult the list of bibliographical references for more details on this exciting and always active research topic.Comment: Tutorial overview of BSS (also presented at SBrT-2001), providing a complete account of the area in early 2000'

A “UNIMODAL ” BLIND EQUALIZATION CRITERION e-mail:

By reexamining some initialization strategies for blind equalizers, we develop an approximation to a common contrast function for equalization which enjoys a “unimodal” property: the solution obtained from the modified criterion is unique to within standard scale factor ambiguity, and moreover yields a perfect equalizer whenever such is attainable. In the more realistic situation in which perfect equalization is unattainable, the modified criterion yields a good approximation which is quantified herein, and is therefore justified as an initialization strategy. An on-line interpretation leads to an adaptive Volterra filter followed by a tensor product approximation which furnishes the coefficients of a linear equalizer, and ties in with some earlier work on equalizer design restricted to finite data sets and Volterra kernel approximations.