A Hierarchical Bayesian Model for Frame Representation

Chaâri, Lotfi; Pesquet, Jean-Christophe; Tourneret, Jean-Yves; Ciuciu, Philippe; Benazza-Benyahia, Amel

doi:10.1109/tsp.2010.2055562

Cited by 30 publications

(22 citation statements)

References 59 publications

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“…Other common choices can be found for instance in [3,4]. Moreover, Ψ(V·) is related to some prior knowledge one can have about x, and V ∈ R M×N is a linear transform that can describe, for example, a frame analysis [5] or a discrete gradient operator [6]. Within a Bayesian framework, it is related to a prior distribution of density p(x) whose logarithm is given by log p(x) = −Ψ(Vx).…”

Section: Introductionmentioning

confidence: 99%

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

Marnissi

Chouzenoux

Benazza-Benyahia

et al. 2018

Entropy

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Abstract:In this paper, we are interested in Bayesian inverse problems where either the data fidelity term or the prior distribution is Gaussian or driven from a hierarchical Gaussian model. Generally, Markov chain Monte Carlo (MCMC) algorithms allow us to generate sets of samples that are employed to infer some relevant parameters of the underlying distributions. However, when the parameter space is high-dimensional, the performance of stochastic sampling algorithms is very sensitive to existing dependencies between parameters. In particular, this problem arises when one aims to sample from a high-dimensional Gaussian distribution whose covariance matrix does not present a simple structure. Another challenge is the design of Metropolis-Hastings proposals that make use of information about the local geometry of the target density in order to speed up the convergence and improve mixing properties in the parameter space, while not being too computationally expensive. These two contexts are mainly related to the presence of two heterogeneous sources of dependencies stemming either from the prior or the likelihood in the sense that the related covariance matrices cannot be diagonalized in the same basis. In this work, we address these two issues. Our contribution consists of adding auxiliary variables to the model in order to dissociate the two sources of dependencies. In the new augmented space, only one source of correlation remains directly related to the target parameters, the other sources of correlations being captured by the auxiliary variables. Experiments are conducted on two practical image restoration problems-namely the recovery of multichannel blurred images embedded in Gaussian noise and the recovery of signal corrupted by a mixed Gaussian noise. Experimental results indicate that adding the proposed auxiliary variables makes the sampling problem simpler since the new conditional distribution no longer contains highly heterogeneous correlations. Thus, the computational cost of each iteration of the Gibbs sampler is significantly reduced while ensuring good mixing properties.

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Section: Introductionmentioning

confidence: 99%

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

Marnissi

Chouzenoux

Benazza-Benyahia

et al. 2018

Entropy

Self Cite

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“…This also raises the question of the computation of the proximity operators associated with the different functions involved in the criterion. Various strategies were proposed in order to address the first question [26,27,28,29,30], but the computational cost of these methods is often high, especially when several regularization parameters have to be set. Alternatively, it has been recognized for a long time that incorporating constraints directly on the solutions [31,32,33,34,35] instead of considering regularized functions may often facilitate the choice of the involved parameters.…”

Section: Introductionmentioning

confidence: 99%

Epigraphical projection and proximal tools for solving constrained convex optimization problems

Chierchia

Pustelnik

Pesquet

et al. 2014

SIViP

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We propose a proximal approach to deal with convex optimization problems involving nonlinear constraints. A large family of such constraints, proven to be effective in the solution of inverse problems, can be expressed as the lower level set of a sum of convex functions evaluated over different, but possibly overlapping, blocks of the signal. For this class of constraints, the associated projection operator generally does not have a closed form. We circumvent this difficulty by splitting the lower level set into as many epigraphs as functions involved in the sum. A closed half-space constraint is also enforced, in order to limit the sum of the introduced epigraphical variables to the upper bound of the original lower level set.In this paper, we focus on a family of constraints involving linear transforms of ℓ1,p balls. Our main theoretical contribution is to provide closed form expressions of the epigraphical projections associated with the Euclidean norm (p = 2) and the sup norm (p = +∞). The proposed approach is validated in the context of image restoration with missing samples, by making use of TV-like constraints. Experiments show that our method leads to significant improvements in term of convergence speed over existing algorithms for solving similar constrained problems.

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“…sparse and clustered. In this subsection, both sparsity and cluster prior are simultaneously modeled through a "spike-and-slab" prior model, also called Bernoulli-Gaussian process [23,24,25], which has been widely used as a sparse promoting prior [26,27,28,29].…”

Section: A Priori Model On Sparsity and Clustermentioning

confidence: 99%

Bayesian compressive sensing for cluster structured sparse signals

et al. 2012

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In traditional framework of Compressive Sensing (CS), only sparse prior on the property of signals in time or frequency domain is adopted to guarantee the exact inverse recovery. Other than sparse prior, structures on the sparse pattern of the signal have also been used as an additional prior, called modelbased compressive sensing, such as clustered structure and tree structure on wavelet coefficients. In this paper, the cluster structured sparse signals are investigated. Under the framework of Bayesian Compressive Sensing, a hierarchical Bayesian model is employed to model both the sparse prior and cluster prior, then Markov Chain Monte Carlo (MCMC) sampling is implemented for the inference. Unlike the state-of-the-art algorithms which are also taking into account the cluster prior, the proposed algorithm solves the inverse problem automatically -prior information on the number of clusters and the size of each cluster is unknown. The experimental results show that the proposed algorithm outperforms many state-of-the-art algorithms.

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A Hierarchical Bayesian Model for Frame Representation

Cited by 30 publications

References 59 publications

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

Epigraphical projection and proximal tools for solving constrained convex optimization problems

Bayesian compressive sensing for cluster structured sparse signals

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