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.

In this letter, we propose a new algorithm for estimating sparse nonnegative sources from a set of noisy linear mixtures. In particular, we consider difficult situations with high noise levels and more sources than sensors (underdetermined case). We show that when sources are very sparse in time and overlapped at some locations, they can be recovered even with very low signal-to-noise ratio, and by using many fewer sensors than sources. A theoretical analysis based on Bayesian estimation tools is included showing strong connections with algorithms in related areas of research such as ICA, NMF, FOCUSS, and sparse representation of data with overcomplete dictionaries. Our algorithm uses a Bayesian approach by modeling sparse signals through mixed-state random variables. This new model for priors imposes ℓ 0 norm-based sparsity. We start our analysis for the case of nonoverlapped sources (1-sparse), which allows us to simplify the search of the posterior maximum avoiding a combinatorial search. General algorithms for overlapped cases, such as 2-sparse and k -sparse sources, are derived by using the algorithm for 1-sparse signals recursively. Additionally, a combination of our MAP algorithm with the NN-KSVD algorithm is proposed for estimating the mixing matrix and the sources simultaneously in a real blind fashion. A complete set of simulation results is included showing the performance of our algorithm.