Lieven De Lathauwer

Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis

The widespread use of multisensor technology and the emergence of big data sets have highlighted the limitations of standard flat-view matrix models and the necessity to move toward more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift toward models that are essentially polynomial, the uniqueness of which, unlike the matrix methods, is guaranteed under very mild and natural conditions. Benefiting from the power of multilinear algebra as their mathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints which match data properties and extract more general latent components in the data than matrix-based methods.

An introduction to independent component analysis

This paper is an introduction to the concept of independent component analysis (ICA) which has recently been developed in the area of signal processing. ICA is a variant of principal component analysis (PCA) in which the components are assumed to be mutually statistically independent instead of merely uncorrelated. The stronger condition allows one to remove the rotational invariance of PCA, i.e. ICA provides a meaningful unique bilinear decomposition of two‐way data that can be considered as a linear mixture of a number of independent source signals. The discipline of multilinear algebra offers some means to solve the ICA problem. In this paper we briefly discuss four orthogonal tensor decompositions that can be interpreted in terms of higher‐order generalizations of the symmetric eigenvalue decomposition. Copyright

Optimization-Based Algorithms for Tensor Decompositions: Canonical Polyadic Decomposition, Decomposition in Rank-

The canonical polyadic and rank-

(L_r,L_r,1)

(L_r,L_r,1)

Terms, and a New Generalization

block term decomposition (CPD and BTD, respectively) are two closely related tensor decompositions. The CPD and, recently, BTD are important tools in psychometrics, chemometrics, neuroscience, and signal processing. We present a decomposition that generalizes these two and develop algorithms for its computation. Among these algorithms are alternating least squares schemes, several general unconstrained optimization techniques, and matrix-free nonlinear least squares methods. In the latter we exploit the structure of the Jacobians Gramian to reduce computational and memory cost. Combined with an effective preconditioner, numerical experiments confirm that these methods are among the most efficient and robust currently available for computing the CPD, rank-

On the Uniqueness of the Canonical Polyadic Decomposition of Third-Order Tensors---Part II: Uniqueness of the Overall Decomposition

(L_r,L_r,1)

Structured Data Fusion

BTD, and their generalized decomposition.

Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices

Canonical polyadic (also known as Candecomp/Parafac) decomposition (CPD) of a higher-order tensor is decomposition into a minimal number of rank-

UNCONSTRAINED OPTIMIZATION OF REAL FUNCTIONS IN COMPLEX VARIABLES

1

On the uniqueness of the canonical polyadic decomposition of third-order tensors - Part I : Basic results and uniqueness of one factor matrix

tensors. In Part I, we gave an overview of existing results concerning uniqueness and presented new, relaxed, conditions that guarantee uniqueness of one factor matrix. In Part II we use these results for establishing overall CPD uniqueness in cases where none of the factor matrices has full column rank. We obtain uniqueness conditions involving Khatri--Rao products of compound matrices and Kruskal-type conditions. We consider both deterministic and generic uniqueness. We also discuss uniqueness of INDSCAL and other constrained polyadic decompositions.