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Abstract: This paper addresses the problem of identifying a lower dimensional spacewhere observed data can be sparsely represented. This under-complete dictionarylearning task can be formulated as a blind separation problem of sparse sourceslinearly mixed with an unknown orthogonal mixing matrix. This issue isformulated in a Bayesian framework. First, the unknown sparse sources aremodeled as Bernoulli-Gaussian processes. To promote sparsity, a weightedmixture of an atom at zero and a Gaussian distribution is proposed as priordistribution for the unobserved sources. A non-informative prior distributiondefined on an appropriate Stiefel manifold is elected for the mixing matrix.The Bayesian inference on the unknown parameters is conducted using a Markovchain Monte Carlo MCMC method. A partially collapsed Gibbs sampler isdesigned to generate samples asymptotically distributed according to the jointposterior distribution of the unknown model parameters and hyperparameters.These samples are then used to approximate the joint maximum a posterioriestimator of the sources and mixing matrix. Simulations conducted on syntheticdata are reported to illustrate the performance of the method for recoveringsparse representations. An application to sparse coding on under-completedictionary is finally investigated.

Author: Nicolas Dobigeon, Jean-Yves Tourneret


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