A Multilinear Singular Value Decomposition
SIAM Journal on Matrix Analysis and Applications
On the Best Rank-1 and Rank-(R1,R2,. . .,RN) Approximation of Higher-Order Tensors
SIAM Journal on Matrix Analysis and Applications
The ubiquitous Kronecker product
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. III: linear algebra
Atomic Decomposition by Basis Pursuit
SIAM Review
Approximation with Kronecker Products
Approximation with Kronecker Products
SIAM Journal on Scientific Computing
Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time
SIAM Journal on Matrix Analysis and Applications
Tensor Decompositions and Applications
SIAM Review
Practical compressive sensing of large images
DSP'09 Proceedings of the 16th international conference on Digital Signal Processing
Block-sparse signals: uncertainty relations and efficient recovery
IEEE Transactions on Signal Processing
Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation
SIAM Journal on Scientific Computing
A Theory for Sampling Signals From a Union of Subspaces
IEEE Transactions on Signal Processing
Sparse representations in unions of bases
IEEE Transactions on Information Theory
Greed is good: algorithmic results for sparse approximation
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit
IEEE Transactions on Information Theory
Kronecker product approximation for preconditioning in three-dimensional imaging applications
IEEE Transactions on Image Processing
Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries
IEEE Transactions on Image Processing
Sparsity and Morphological Diversity in Blind Source Separation
IEEE Transactions on Image Processing
IEEE Transactions on Image Processing
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Recently there has been great interest in sparse representations of signals under the assumption that signals data sets can be well approximated by a linear combination of few elements of a known basis dictionary. Many algorithms have been developed to find such representations for one-dimensional signals vectors, which requires finding the sparsest solution of an underdetermined linear system of algebraic equations. In this letter, we generalize the theory of sparse representations of vectors to multiway arrays tensors-signals with a multidimensional structure-by using the Tucker model. Thus, the problem is reduced to solving a large-scale underdetermined linear system of equations possessing a Kronecker structure, for which we have developed a greedy algorithm, 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 not only has a considerably lower complexity but is also more precise than the classic OMP algorithm. Moreover, our algorithms can be used for very large-scale problems, which are intractable 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 data sets, in particular, for 2D compressive imaging CI and 3D hyperspectral CI, and we show examples with real-world multidimensional signals.