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SOLVING A LOWRANK FACTORIZATION MODEL FOR MATRIX COMPLETION BY A NONLINEAR SUCCESSIVE OVERRELAXATION ALGORITHM
"... Abstract. The matrix completion problem is to recover a lowrank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclearnorm minimization which requires computing singular value decompositions – a task that is increasingly costly as matrix sizes an ..."
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Abstract. The matrix completion problem is to recover a lowrank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclearnorm minimization which requires computing singular value decompositions – a task that is increasingly costly as matrix sizes and ranks increase. To improve the capacity of solving largescale problems, we propose a lowrank factorization model and construct a nonlinear successive overrelaxation (SOR) algorithm that only requires solving a linear least squares problem per iteration. Convergence of this nonlinear SOR algorithm is analyzed. Numerical results show that the algorithm can reliably solve a wide range of problems at a speed at least several times faster than many nuclearnorm minimization algorithms. Key words. Matrix Completion, alternating minimization, nonlinear GS method, nonlinear SOR method AMS subject classifications. 65K05, 90C06, 93C41, 68Q32
Multiarray Signal Processing: Tensor decomposition meets compressed sensing
, 2009
"... We discuss how recently discovered techniques and tools from compressed sensing can be used in tensor decompositions, with a view towards modeling signals from multiple arrays of multiple sensors. We show that with appropriate bounds on coherence, one could always guarantee the existence and uniquen ..."
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We discuss how recently discovered techniques and tools from compressed sensing can be used in tensor decompositions, with a view towards modeling signals from multiple arrays of multiple sensors. We show that with appropriate bounds on coherence, one could always guarantee the existence and uniqueness of a best rankr approximation of a tensor. In particular, we obtain a computationally feasible variant of Kruskal’s uniqueness condition with coherence as a proxy for krank. We treat sparsest recovery and lowestrank recovery problems in a uniform fashion by considering Schatten and nuclear norms of tensors of arbitrary order and dictionaries that comprise a continuum of uncountably many atoms.
Multiarray Signal Processing: Tensor decomposition meets compressed sensing
"... We discuss how recently discovered techniques and tools from compressed sensing can be used in tensor decompositions, with a view towards modeling signals from multiple arrays of multiple sensors. We show that with appropriate bounds on a measure of separation between radiating sources called cohe ..."
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We discuss how recently discovered techniques and tools from compressed sensing can be used in tensor decompositions, with a view towards modeling signals from multiple arrays of multiple sensors. We show that with appropriate bounds on a measure of separation between radiating sources called coherence, one could always guarantee the existence and uniqueness of a best rankr approximation of the tensor representing the signal. We also deduce a computationally feasible variant of Kruskal’s uniqueness condition, where the coherence appears as a proxy for krank. Problems of sparsest recovery with an infinite continuous dictionary, lowestrank tensor representation, and blind source separation are treated in a uniform fashion. The decomposition of the measurement tensor leads to simultaneous localization and extraction of radiating sources, in an entirely deterministic manner. Résumé Traitement du signal multiantenne: les décompositions tensorielles rejoignent l’échantillonnage compressé. Nous décrivons comment les techniques et outils d’échantillonnage compresse ́ récemment découverts peuvent être utilisés dans les décompositions tensorielles, avec pour illustration une modélisation des signaux provenant de plusieurs antennes multicapteurs. Nous montrons qu’en posant des bornes appropriées sur une certaine mesure de séparation entre les sources rayonnantes (appelée cohérence dans le jargon de l’échantillonnage compressé), on pouvait toujours