SummaryThe widespread use of multi-sensor technology and the emergence of big datasets has highlighted the limitations of standard flat-view matrix models and the necessity to move towards more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift towards models that are essentially polynomial and whose uniqueness, 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 that match data properties, and to find more general latent components in the data than matrix-based methods. A comprehensive introduction to tensor decompositions is provided from a signal processing perspective, starting from the algebraic foundations, via basic Canonical Polyadic and Tucker models, through to advanced cause-effect and multi-view data analysis schemes. We show that tensor decompositions enable natural generalizations of some commonly used signal processing paradigms, such as canonical correlation and subspace techniques, signal separation, linear regression, feature extraction and classification. We also cover computational aspects, and point out how ideas from compressed sensing and scientific computing may be used for addressing the otherwise unmanageable storage and manipulation problems associated with big datasets. The concepts are supported by illustrative real world case studies illuminating the benefits of the tensor framework, as efficient and promising tools for modern signal processing, data analysis and machine learning applications; these benefits also extend to vector/matrix data through tensorization.

Tractography uses diffusion MRI to estimate the trajectory and cortical projection zones of white matter fascicles in the living human brain. There are many different tractography algorithms and each requires the user to set several parameters, such as curvature threshold. Choosing a single algorithm with specific parameters poses two challenges. First, different algorithms and parameter values produce different results. Second, the optimal choice of algorithm and parameter value may differ between different white matter regions or different fascicles, subjects, and acquisition parameters. We propose using ensemble methods to reduce algorithm and parameter dependencies. To do so we separate the processes of fascicle generation and evaluation. Specifically, we analyze the value of creating optimized connectomes by systematically combining candidate streamlines from an ensemble of algorithms (deterministic and probabilistic) and systematically varying parameters (curvature and stopping criterion). The ensemble approach leads to optimized connectomes that provide better cross-validated prediction error of the diffusion MRI data than optimized connectomes generated using a single-algorithm or parameter set. Furthermore, the ensemble approach produces connectomes that contain both short- and long-range fascicles, whereas single-parameter connectomes are biased towards one or the other. In summary, a systematic ensemble tractography approach can produce connectomes that are superior to standard single parameter estimates both for predicting the diffusion measurements and estimating white matter fascicles.

Recently, there is a great interest in sparse representations of signals under the assumption that signals (datasets) can be well approximated by a linear combination of few elements of a known basis (dictionary). Many algorithms have been developed to find such kind of representations for the case of one-dimensional signals (vectors) which involves to find the sparsest solution of an underdetermined linear system of algebraic equations. In this paper, we generalize the theory of sparse representations of vectors to multiway arrays (tensors), i.e. signals with a multidimensional structure, by using the Tucker model. Thus, the problem is reduced to solve a large-scale underdetermined linear system of equations possessing a Kronecker structure, for which we have developed a greedy algorithm called Kronecker-OMP as a generalization of the classical Orthogonal Matching Pursuit (OMP) algorithm for vectors. We also introduce the concept of multiway block-sparse representation of N -way arrays and develop a new greedy algorithm that exploits not only the Kronecker structure but also block-sparsity. This allows us to derive a very fast and memory efficient algorithm called N-BOMP (N -way Block OMP). We theoretically demonstrate that, under the block-sparsity assumption, our N-BOMP algorithm has not only a considerably lower complexity but it is also more precise than the classical OMP algorithm. Moreover, our algorithms can be used for very large-scale problems which are intractable by using standard approaches. We provide several simulations illustrating our results and comparing our algorithms to classical algorithms such as OMP and BP (Basis Pursuit) algorithms. We also apply the N-BOMP algorithm as a fast solution for the Compressed Sensing (CS) problem with large-scale datasets, in particular for 2D Compressive Imaging (CI) and 3D Hyperspectral CI and we show examples with real world multidimensional signals.