@MISC{_1computing, author = {}, title = {1 Computing Sparse Representations of Multidimensional Signals Using Kronecker Bases}, year = {} }

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Abstract

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).