Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimal-sparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.

A full-rank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easily-verifiable conditions under which optimally-sparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several well-known signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical

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—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.

Abstract—Parallel factor (PARAFAC) analysis is an extension of low-rank matrix decomposition to higher way arrays, also referred to as tensors. It decomposes a given array in a sum of multilinear terms, analogous to the familiar bilinear vector outer products that appear in matrix decomposition. PARAFAC analysis generalizes and unifies common array processing models, like joint diagonalization and ESPRIT; it has found numerous applications from blind multiuser detection and multidimensional harmonic retrieval, to clustering and nuclear magnetic resonance. The prevailing fitting algorithm in all these applications is based on (alternating) least squares, which is optimal for Gaussian noise. In many cases, however, measurement errors are far from being Gaussian. In this paper, we develop two iterative algorithms for the least absolute error fitting of general multilinear models. The first is based on efficient interior point methods for linear programming, employed in an alternating fashion. The second is based on a weighted median filtering iteration, which is particularly appealing from a simplicity viewpoint. Both are guaranteed to converge in terms of absolute error. Performance is illustrated by means of simulations, and compared to the pertinent Cramér–Rao bounds (CRBs). Index Terms—Array signal processing, non-Gaussian noise, parallel factor analysis, robust model fitting. I.

In many signal processing applications of linear algebra tools, the signal part of a postulated model lies in a so-called signal subspace, while the parameters of interest are in one-to-one correspondence with a certain basis of this subspace. The signal subspace can often be reliably estimated from measured data, but the particular basis of interest cannot be identified without additional problem-specific structure. This is a manifestation of rotational indeterminacy, i.e., non-uniqueness of low-rank matrix decomposition. The situation is very different for three- or higher-way arrays, i.e., arrays indexed by three or more independent variables, for which low-rank decomposition is unique under mild conditions. This has fundamental implications for DSP problems which deal with such data. This paper provides a brief tour of the basic elements of this theory, along with many examples of application in problems of current interest in the signal processing community. Keywords: Three-way analysis, low-rank decomposition, parallel factor analysis (PARAFAC), canonical decomposition (CAN-

Abstract—This paper considers the problem of blind localization and tracking of multiple frequency-hopped spread-spectrum signals using a uniform linear antenna array without knowledge of hopping patterns or directions of arrival. As a preprocessing step, we propose to identify a hop-free subset of data by discarding high-entropy spectral slices from the spectrogram. High-resolution localization is then achieved via either quadrilinear regression of four-way data generated by capitalizing on both spatial and temporal shift invariance or a new maximum likelihood (ML)-based two-dimensional (2-D) harmonic retrieval algorithm. The latter option achieves the best-known model identifiability bound while remaining close to the Cramér–Rao bound even at low signal-to-noise ratios (SNRs). Following beamforming using the recovered directions, a dynamic programming approach is developed for joint ML estimation of signal frequencies and hop instants in single-user tracking. The efficacy of the proposed algorithms is illustrated in pertinent simulations. Index Terms—Array signal processing, direction-of-arrival (DOA) estimation, frequency estimation, frequency hopping, harmonic analysis. I.

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In the context of multiuser detection for the DS-CDMA uplink, out-of-cell interference is usually treated as Gaussian noise, possibly mitigated by overlaying a long randomcellcodeontopofsymbolspreading.Different cells use statistically independent long codes, thereby providing means for statistical out-of-cell interference suppression. When the total number of (in-cell plus outof-cell) users is less than the spreading gain, subspace identification techniques are applicable. If the base station is equipped with multiple antennas, then completely blind identification is possible via three-dimensional low-rank decomposition. This works with more users than spreading and antennas, but a purely algebraic solution is missing. In this paper, we develop an algebraic solution under the premise that the codes of the in-cell users are known. The codes of out-of-cell users and all array steering vectors are unknown. In this pragmatic scenario, we show that in addition to algebraic solution, better identifiability is possible. Our approach yields the best known identifiability result for three-dimensional low-rank decomposition when one of the three component matrices is partially known, albeit noninvertible. Simulations show that the proposed identification algorithm remains close to the pertinent asymptotic (symbol-independent) Cramér-Rao bound, which is also derived herein.

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