Signal Processing Computations Using The Generalized Singular Value Decomposition

Abstract

]. 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 to two tenso rs A and B rather than to a singl e tenso r A. Typically one tenso r (call ed A) contains the valuable information, whereas the other (call ed B) contains information that is irrel evant and hence shoul d be rejected. This is the class of probl ems that are tackl ed in this paper. It turns out that there are several ways to exten d the relev ant matri x concepts, and that their exten sions are not trivi al.

Unambiguous discrimination of mixed states

“…As such, our results can be applied to the task of achieving an UD of two entangled states of a composite system given access to only one of the subsystems. It is also worth noting that our lower bound is obtained by making implicit use of the CS decomposition, which constitutes a powerful tool in both modern linear algebra and classical signal analysis [12]. To our knowledge, this is the first application of the CS decomposition in quantum signal analysis.…”

Hierarchical Linear Discriminant Analysis for Beamforming

“…m < n. The basic algorithm for h-LDA is primarily based on the generalized singular value decomposition (GSVD) framework, which has its foundations in the original LDA solution via the GSVD [11,22], LDA/GSVD. We also point out that the GSVD algorithm is a familiar method to the signal processing community, particularly for direction-of-arrival (DOA) estimation [23]. In order to describe the h-LDA algorithm, let us define the "square-root" factors, H ws , H bs , H h w , and H b of S ws , S bs , S h w , and S b , respectively, as n+s) , and…”

Computing the generalized singular values/vectors of large sparse or structured matrix pairs

“…In the singular case, {A, B } has a infinite generalized singular value while all the singular values of AB † are finite. This problem was also pointed out in [14,22,30]. We compare the accuracy of the computed generalized singular values and vectors that can be achieved by the "exact" method and the Modified Lanczos Algorithm, the results are plotted in Fig.…”

“…The generalized singular value decomposition (GSVD) is one of the essential numerical linear algebra tools in signal processing and system identification [21,22]. It is a generalization of the familiar singular value decomposition (SVD) to the case of matrix pairs.…”

“…These algorithms are geared to the computation of the complete decomposition and they do not take into account of the sparsity or structure of the matrix pair {A, B }. In some applications, for example, the generalized total least squares problem [27] and direction-of-arrival estimation with colored noise [22], the complete decomposition is not necessary: only a few of the generalized singular vectors corresponding to the smallest or largest generalized singular values are needed. In those applications, the matrix pair {A, B } can be sparse or structured, for example, A and B are toeplitz matrices.…”