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An affine scaling methodology for best basis selection
 IEEE Trans. Signal Processing
, 1999
"... Abstract — A methodology is developed to derive algorithms for optimal basis selection by minimizing diversity measures proposed by Wickerhauser and Donoho. These measures include the pnormlike (`(p 1)) diversity measures and the Gaussian and Shannon entropies. The algorithm development methodolog ..."
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Cited by 79 (11 self)
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Abstract — A methodology is developed to derive algorithms for optimal basis selection by minimizing diversity measures proposed by Wickerhauser and Donoho. These measures include the pnormlike (`(p 1)) diversity measures and the Gaussian and Shannon entropies. The algorithm development methodology uses a factored representation for the gradient and involves successive relaxation of the Lagrangian necessary condition. This yields algorithms that are intimately related to the Affine Scaling Transformation (AST) based methods commonly employed by the interior point approach to nonlinear optimization. The algorithms minimizing the `(p 1) diversity measures are equivalent to a recently developed class of algorithms called FOCal Underdetermined System Solver (FOCUSS). The general nature of the methodology provides a systematic approach for deriving this class of algorithms and a natural mechanism for extending them. It also facilitates a better understanding of the convergence behavior and a strengthening of the convergence results. The Gaussian entropy minimization algorithm is shown to be equivalent to a wellbehaved p =0normlike optimization algorithm. Computer experiments demonstrate that the pnormlike and the Gaussian entropy algorithms perform well, converging to sparse solutions. The Shannon entropy algorithm produces solutions that are concentrated but are shown to not converge to a fully sparse solution. I.
An improved FOCUSSbased learning algorithm for solving sparse linear inverse problems
 in Conf. Record of the ThirtyFifth Asilomar Conf. on Signals, Systems and Computers
, 2001
"... We develop an improved algorithm for solving blind sparse linear inverse problems where both the dictionary (possibly overcomplete) and the sources are unknown. The algorithm is derived in the Bayesian framework by the maximum a posteriori method, with the choice of prior distribution restricted to ..."
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Cited by 19 (2 self)
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We develop an improved algorithm for solving blind sparse linear inverse problems where both the dictionary (possibly overcomplete) and the sources are unknown. The algorithm is derived in the Bayesian framework by the maximum a posteriori method, with the choice of prior distribution restricted to the class of concave/Schurconcave functions, which has been shown previously to be a sufficient condition for sparse solutions. This formulation leads to a constrained and regularized minimization problem which can be solved in part using the FOCUSS (Focal Underdetermined System Solver) algorithm for vector selection. We introduce three key improvements in the algorithm: an efficient way of adjusting the regularization parameter, column normalization that restricts the learned dictionary, and reinitialization to escape from local optima. Experiments were performed using synthetic data with matrix sizes up to 64x128, and the algorithm is shown to solve the blind identification problem, recovering both the dictionary and the sparse sources. The improved algorithm is shown to be much more accurate than the original FOCUSSDictionary Learning algorithm when using large matrices. We also test our algorithm on natural images, and show that a learned overcomplete representation can encode the data more efficiently than a complete basis at the same level of accuracy. 1
Methodes De Separation De Sources Dans Le Cas
"... In this contribution, the underdetermined blind source separation problem is addressed. We recall some known identi ability results, and present various methods for the identi cation of the mixture matrix, and the extraction of the sources. Finally computer simulations illustrate the identi catio ..."
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In this contribution, the underdetermined blind source separation problem is addressed. We recall some known identi ability results, and present various methods for the identi cation of the mixture matrix, and the extraction of the sources. Finally computer simulations illustrate the identi cation algorithms.
Bayesian Modelling of Music: Algorithmic Advances and . . .
, 2005
"... In order to perform many signal processing tasks such as classification, pattern recognition and coding, it is helpful to specify a signal model in terms of meaningful signal structures. In general, designing such a model is complicated and for many signals it is not feasible to specify the appropri ..."
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In order to perform many signal processing tasks such as classification, pattern recognition and coding, it is helpful to specify a signal model in terms of meaningful signal structures. In general, designing such a model is complicated and for many signals it is not feasible to specify the appropriate structure. Adaptive models overcome this problem by learning structures from a set of signals. Such adaptive models need to be general enough, so that they can represent relevant structures. However, more general models often require additional constraints to guide the learning procedure. In this thesis