Mikael Sorensen

Coupled Canonical Polyadic Decompositions and (Coupled) Decompositions in Multilinear Rank-

The coupled canonical polyadic decomposition (CPD) is an emerging tool for the joint analysis of multiple data sets in signal processing and statistics. Despite their importance, linear algebra based algorithms for coupled CPDs have not yet been developed. In this paper, we first explain how to obtain a coupled CPD from one of the individual CPDs. Next, we present an algorithm that directly takes the coupling between several CPDs into account. We extend the methods to single and coupled decompositions in multilinear rank-

(L_{r,n},L_{r,n},1)

(L_{{r,n}},L_{{r,n}},1)

Terms---Part II: Algorithms

terms. Finally, numerical experiments demonstrate that linear algebra based algorithms can provide good results at a reasonable computational cost.

Blind Signal Separation via Tensor Decomposition With Vandermonde Factor: Canonical Polyadic Decomposition

Several problems in signal processing have been formulated in terms of the Canonical Polyadic Decomposition of a higher-order tensor with one or more Vandermonde constrained factor matrices. We first propose new, relaxed uniqueness conditions. We explain that, under these conditions, the number of components may simply be estimated as the rank of a matrix. We propose an efficient algorithm for the computation of the factors that only resorts to basic linear algebra. We demonstrate the use of the results for various applications in wireless communication and array processing.

Canonical Polyadic Decomposition with a columnwise orthonormal factor matrix

Canonical polyadic decomposition (CPD) of a higher-order tensor is an important tool in mathematical engineering. In many applications at least one of the matrix factors is constrained to be columnwise orthonormal. We first derive a relaxed condition that guarantees uniqueness of the CPD under this constraint. Second, we give a simple proof of the existence of the optimal low-rank approximation of a tensor in the case that a factor matrix is columnwise orthonormal. Third, we derive numerical algorithms for the computation of the constrained CPD. In particular, orthogonality-constrained versions of the CPD methods based on simultaneous matrix diagonalization and alternating least squares are presented. Numerical experiments are reported.

Decomposing tensors with structured matrix factors reduces to rank-1 approximations

Tensor decompositions permit to estimate in a deterministic way the parameters in a multi-linear model. Applications have been already pointed out in antenna array processing and digital communications, among others, and are extremely attractive provided some diversity at the receiver is available. As opposed to the widely used ALS algorithm, non-iterative algorithms are proposed in this paper to compute the required tensor decomposition into a sum of rank-1 terms, when some factor matrices enjoy some structure, such as block-Hankel, triangular, band, etc.

New Uniqueness Conditions for the Canonical Polyadic Decomposition of Third-Order Tensors

The uniqueness properties of the canonical polyadic decomposition (CPD) of higher-order tensors make it an attractive tool for signal separation. However, CPD uniqueness is not yet fully understood. In this paper, we first present a new uniqueness condition for a polyadic decomposition (PD) where one of the factor matrices is assumed to be known. We also show that this result can be used to obtain a new overall uniqueness condition for the CPD. In signal processing the CPD factor matrices are often constrained. Building on the preceding results, we provide a new uniqueness condition for a CPD with a columnwise orthonormal factor matrix, representing uncorrelated signals. We also obtain a new uniqueness condition for a CPD with a partial Hermitian symmetry, useful for tensors in which covariance matrices are stacked, which are common in statistical signal processing. We explain that such constraints can lead to more relaxed uniqueness conditions. Finally, we provide an inexpensive algorithm for computing a...

Coupled tensor decompositions for applications in array signal processing

For the case of a single colocated receive antenna array and additional linear diversity (e.g. oversampling or polarization), tensor decomposition based signal separation is now well-established. For increasing the spatial diversity of communication systems, the use of several widely separated colocated antenna arrays has been suggested. However, for such problems no algebraic framework has been proposed. We explain that recently developed coupled tensor decompositions provide such a framework. In particular, we explain that the use of several widely separated colocated antenna arrays may lead to improved identifiability results.

Multidimensional Harmonic Retrieval via Coupled Canonical Polyadic Decomposition—Part I: Model and Identifiability

Multidimensional Harmonic Retrieval (MHR) is a fundamental problem in signal processing. We make a connection with coupled Canonical Polyadic Decomposition (CPD), which allows us to better exploit the rich MHR structure than existing approaches in the derivation of uniqueness results. We discuss both deterministic and generic conditions. We obtain a deterministic condition that is both necessary and sufficient but which may be difficult to check in practice. We derive mild deterministic relaxations that are easy to verify. We also discuss the variant in which the generators have unit norm. We narrow the transition zone between generic uniqueness and generic non-uniqueness to two values of the number of harmonics. We explain differences with one-dimensional HR.

Parafac with orthogonality in one mode and applications in DS-CDMA systems

Blind deterministic receivers for DS-CDMA systems based on the PARAFAC model have been proposed in several papers since their conception in. In many cases, the transmitted signals can be considered uncorrelated. Hence, we develop PARAFAC receivers for uncorrelated signals. We introduce several numerical algorithms for orthogonality constrained PARAFAC on which receivers for uncorrelated signals can be based. Simulation results show an increase in performance when the PARAFAC receiver takes the uncorrelatedness of the transmitted signals into account.