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Aiding Dictionary Learning Through Multi-Parametric Sparse Representation

1
Department of Automatic Control and Computers, University Politehnica of Bucharest, 313 Spl. Independenței, 060042 Bucharest, Romania
2
The Research Institute of the University of Bucharest (ICUB) and Department of Computer Science, University of Bucharest, Bulevardul M. Kogălniceanu 36-46, 050107 Bucharest, Romania
*
Author to whom correspondence should be addressed.
Algorithms 2019, 12(7), 131; https://doi.org/10.3390/a12070131
Received: 20 May 2019 / Revised: 21 June 2019 / Accepted: 25 June 2019 / Published: 28 June 2019
(This article belongs to the Special Issue Dictionary Learning Algorithms and Applications)
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Abstract

The 1 relaxations of the sparse and cosparse representation problems which appear in the dictionary learning procedure are usually solved repeatedly (varying only the parameter vector), thus making them well-suited to a multi-parametric interpretation. The associated constrained optimization problems differ only through an affine term from one iteration to the next (i.e., the problem’s structure remains the same while only the current vector, which is to be (co)sparsely represented, changes). We exploit this fact by providing an explicit, piecewise affine with a polyhedral support, representation of the solution. Consequently, at runtime, the optimal solution (the (co)sparse representation) is obtained through a simple enumeration throughout the non-overlapping regions of the polyhedral partition and the application of an affine law. We show that, for a suitably large number of parameter instances, the explicit approach outperforms the classical implementation. View Full-Text
Keywords: dictionary learning; multi-parametric problem; sparse representation; cosparse representation; cross-polytopic constraint; rank-deficient quadratic cost dictionary learning; multi-parametric problem; sparse representation; cosparse representation; cross-polytopic constraint; rank-deficient quadratic cost
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Stoican, F.; Irofti, P. Aiding Dictionary Learning Through Multi-Parametric Sparse Representation. Algorithms 2019, 12, 131.

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