Journal
IEEE TRANSACTIONS ON SIGNAL PROCESSING
Volume 52, Issue 8, Pages 2153-2164Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TSP.2004.831016
Keywords
basis selection; diversity measures; linear inverse problems; sparse Bayesian learning; sparse representations
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Sparse Bayesian learning (SBL) and specifically relevance vector machines have received much attention in the machine learning literature as a means of achieving parsimonious representations in the context of regression and classification. The methodology relies on a parameterized prior that encourages models with few nonzero weights. In this paper, we adapt SBL to the signal processing problem of basis selection from overcomplete dictionaries, proving, several results About the SBL cost function that elucidate its general behavior and provide solid theoretical justification for this application. Specifically, we have shown that SBL retains a desirable property of the l(0)-norm diversity measure (i.e., the global minimum is achieved at the maximally sparse solution) while often possessing a more limited constellation of local minima. We have also demonstrated that the local minima that do exist are achieved at sparse solutions. Later, we provide a novel interpretation of SBL that gives us valuable insight into why it is successful in producing sparse representations. Finally, we include simulation studies comparing sparse Bayesian learning with Basis Pursuit and the more recent FOCal Underdetermined System Solver (FOCUSS) class of basis selection algorithms. These results indicate that our theoretical insights translate directly into improved performance.
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