Neural Comput - Computing sparse representations of multidimensional signals using Kronecker bases.

Tópicos

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Resumo

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.

Resumo Limpo

recent great interest spars represent signal assumpt signal data set can well approxim linear combin element known basi dictionari mani algorithm develop find represent onedimension signal vector requir find sparsest solut underdetermin linear system algebra equat letter general theori spars represent vector multiway array tensorssign multidimension structurebi use tucker model thus problem reduc solv largescal underdetermin linear system equat possess kroneck structur develop greedi algorithm kroneckeromp general classic orthogon match pursuit omp algorithm vector also introduc concept multiway blockspars represent nway array develop new greedi algorithm exploit kroneck structur also block sparsiti allow us deriv fast memoryeffici algorithm call nbomp nway block omp theoret demonstr blockspars assumpt nbomp algorithm consider lower complex also precis classic omp algorithm moreov algorithm can use largescal problem intract use standard approach provid sever simul illustr result compar algorithm classic algorithm omp bp basi pursuit algorithm also appli nbomp algorithm fast solut compress sens cs problem largescal data set particular d compress imag ci d hyperspectr ci show exampl realworld multidimension signal

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