This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or N -way array. Decompositions of higher-order tensors (i.e., N -way arrays with N ≥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, etc. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decompo- sition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal components analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox and Tensor Toolbox, both for MATLAB, and the Multilinear Engine are examples of software packages for working with tensors.

Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank-1 order-k tensor is the outer product of k non-zero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We

Linear mixtures of independent random variables (the so-called sources) are sometimes referred to as under-determined mixtures (UDM) when the number of sources exceeds the dimension of the observation space. The algorithms proposed are able to identify algebraically a UDM using the second characteristic function (c.f.) of the observations, without any need of sparsity assumption on sources. In fact, by taking higher order derivatives of the multivariate c.f. core equation, the blind identification problem is shown to reduce to a tensor decomposition. With only two sensors, the first algorithm only needs a SVD. With a larger number of sensors, the second algorithm executes an alternating least squares (ALS) algorithm. The joint use of statistics of different orders is possible, and a LS solution can be computed. Identifiability conditions are stated in each of the two cases. Computer simulations eventually demonstrate performances in the absence of sparsity, and emphasize the interest in using jointly derivatives of different orders. r 2005 Elsevier B.V. All rights reserved.

Abstract. The canonical decomposition of higher-order tensors is a key tool in multilinear algebra. First we review the state of the art. Then we show that, under certain conditions, the problem can be rephrased as the simultaneous diagonalization, by equivalence or congruence, of a set of matrices. Necessary and sufficient conditions for the uniqueness of these simultaneous matrix decompositions are derived. In a next step, the problem can be translated into a simultaneous generalized Schur decomposition, with orthogonal unknowns [A.-J. van der Veen and A. Paulraj, IEEE Trans. Signal Process., 44 (1996), pp. 1136–1155]. A first-order perturbation analysis of the simultaneous generalized Schur decomposition is carried out. We discuss some computational techniques (including a new Jacobi algorithm) and illustrate their behavior by means of a number of numerical experiments.

Abstract—CANDECOMP/PARAFAC (CP) analysis is an extension of low-rank matrix decomposition to higher-way arrays, which are also referred to as tensors. CP extends and unifies several array signal processing tools and has found applications ranging from multidimensional harmonic retrieval and angle-carrier estimation to blind multiuser detection. The uniqueness of CP decomposition is not fully understood yet, despite its theoretical and practical significance. Toward this end, we first revisit Kruskal’s Permutation Lemma, which is a cornerstone result in the area, using an accessible basic linear algebra and induction approach. The new proof highlights the nature and limits of the identification process. We then derive two equivalent necessary and sufficient uniqueness conditions for the case where one of the component matrices involved in the decomposition is full column rank. These new conditions explain a curious example provided recently in a previous paper by Sidiropoulos, who showed that Kruskal’s condition is in general sufficient but not necessary for uniqueness and that uniqueness depends on the particular joint pattern of zeros in the (possibly pretransformed) component matrices. As another interesting application of the Permutation Lemma, we derive a similar necessary and sufficient condition for unique bilinear factorization under constant modulus (CM) constraints, thus providing an interesting link to (and unification with) CP. Index Terms—CANDECOMP, constant modulus, identifiablity, PARAFAC, SVD, three-way array analysis, uniqueness. I.

This paper deals with the multiuser medium access problem in the packet radio environment. Under the framework of network diversity multiple access (NDMA), a recently proposed medium access method, a blind collision resolution scheme is proposed employing rotational invariance and factor analysis techniques. The proposed approach (dubbed B-NDMA for Blind NDMA) overcomes the difficulty of orthogonal identification codes required by the original protocol, thereby improving channel utilization and system capacity, while being insensitive to multipath effects and synchronization errors. Performance issues of the proposed technique are addressed both analytically and numerically. Index Terms---Access protocols, blind signal separation, matrix decomposition, packet radio, rotational invariance.

Unlike low-rank matrix decomposition, which is generically nonunique for rank greater than one, low-rank threeand higher dimensional array decomposition is unique, provided that the array rank is lower than a certain bound, and the correct number of components (equal to array rank) is sought in the decomposition. Parallel factor (PARAFAC) analysis is a common name for low-rank decomposition of higher dimensional arrays. This paper develops Cramr--Rao Bound (CRB) results for low-rank decomposition of three- and four-dimensional (3-D and 4-D) arrays, illustrates the behavior of the resulting bounds, and compares alternating least squares algorithms that are commonly used to compute such decompositions with the respective CRBs. Simple-to-check necessary conditions for a unique low-rank decomposition are also provided. Index Terms---Cramr--Rao bound, least squares method, matrix decomposition, multidimensional signal processing. I.

INTRODUCTION Suppose that we are given K complex Hermitian matrices Y k of the form where the k are diagonal and real, and E k represents additive noise. The joint diagonalization problem we consider is, given the Y k , to estimate the common factor A. We assume that all Y k are square d d matrices, and that A is square d d with full rank d. An extension of this problem is, for complex non-Hermitian matrices, where A and B can be different, and the k are diagonal but not necessarily real. Joint diagonalization of either type turns up in several recently proposed blind source separation problems with data models X AS N, where X is the observation matrix, A is the mixing matrix, the rows of S contain the source signals, and N is additive noise. Depending on the assumptions on A and/or S, the following types of algebraic source separation techniques have been proposed: -- Diagonalization of fourth order cumulant matrices, as in JADE [1] where K d and A is considered unitary.

Blind Identification of Under-Determined Mixtures (UDM) is involved in numerous applications, including Multi-Way factor Analysis (MWA) and Signal Processing. In the latter case, the use of High-Order Statistics (HOS) like Cumulants leads to the decomposition of symmetric tensors. Yet, little has been published about rank-revealing decompositions of symmetric tensors. Definitions of rank are discussed, and useful results on Generic Rank are proved, with the help of tools borrowed from Algebraic Geometry. 1.

Abstract—Space-time (ST) coding techniques exploit the spatial diversity afforded by multiple transmit and receive antennas to achieve reliable transmission in scattering-rich environments. ST block codes are capable of realizing full diversity and spatial coding gains at relatively low rates; ST trellis codes can achieve better rate-diversity tradeoffs at the cost of high complexity. On the other hand, V-BLAST supports high rates but has no built-in spatial coding and does not work well with fewer receive than transmit antennas. We propose a novel linear block-coding scheme based on the Khatri-Rao matrix product. The proposed scheme offers flexibility for achieving full-rate or full-diversity, or a desired rate-diversity tradeoff, and it can handle any transmit/receive antenna configuration or signal constellation. The proposed codes are shown to have numerous desirable properties, including guaranteed unique linear decodability, built-in blind channel identifiability, and efficient near-maximum likelihood decoding. Index Terms—Blind channel identifiablilty, fading channels, multi-antenna systems, receive diversity, space-time codes, transmit diversity, wireless communications. I.