The generalized higher order singular value decomposition and the oriented signal-to-signal ratios of pairs of signal tensors and their use in signal processing

Abstr act − Two new gener alizations of tenso r conce pts for signa l proce ssing are prese nted. These gener ali-zatio ns are typic ally relev ant for appli cations where one tenso r consi sts of valua ble measu red data or signa ls, that shoul d be retai ned, while the secon d tenso r conta ins data or information that shoul d be rejec ted. First the highe r order singu lar value decomposition for a singl e tenso r is exten ded to pairs of tenso rs; this is the multi linear equiv alent of the gener alized or quoti ent SVD (GSVD, QSVD) for pairs of matri ces. Next the notio n of orien ted signa l-to-signa l ratio s that was deriv ed for pairs of matri ces is exten ded to pairs of tenso rs. These signa l to signa l ratio s can be linke d to the previ ously defin ed gener alized highe r order singu lar value decomposition. 1 INTRODUCTION In recen t years more and more instances of appli cations occur, where the data have more than two indices and hence are not organ ized in a matri x but in a tenso r, also called a multi way or multi dimensional array. Let us menti on here psych ometrics [16], chemometrics [7, 8, 15, 17] and stati stical signal and image processing [6, 9, 10, 18]. In typical image appli cations, the 4 different indices of a 4 th order image tenso r can be the x, y, t and color axes. A recen t appli cation is the websearch tenso r. Here we will mainl y work in a signal processing, but many concepts can be carri ed over to the other domai ns. Typically the metho ds of matri x theory and relat ed numerical compu tations [1] are no longer adequ ate and valuable for tenso rs. Therefore a number of studi es [10-1 4, 19-20 ] have been perfo rmed to exten d some matri x concepts to tenso rs, like the higher order singu lar value decom position or the canon ical decom position of a tenso r. Also the use of tenso rs for findi ng indep endent compo nents [9] in signals is a topic of current interest. Many appli cations occur where the measu red data lead …

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