C. Elvira, P. Chainais, and N. Dobigeon, Democratic prior for antisparse coding, Proc. IEEE-SP Workshop Stat. and Signal Process. (SSP), 2016.
DOI : 10.1109/ssp.2016.7551813
URL : https://hal.archives-ouvertes.fr/hal-01500503

E. J. Candès and M. B. Wakin, An Introduction To Compressive Sampling, IEEE Signal Processing Magazine, vol.25, issue.2, pp.21-30, 2008.
DOI : 10.1109/MSP.2007.914731

M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok, Introduction to compressed sensing, Compressed Sensing: Theory and Applications, pp.1-64, 2012.
DOI : 10.1017/CBO9780511794308.002

R. G. Baraniuk, Compressive sensing, 2008 42nd Annual Conference on Information Sciences and Systems, pp.118-121, 2007.
DOI : 10.1109/CISS.2008.4558479
URL : https://hal.archives-ouvertes.fr/hal-00452261

I. Tosic and P. Frossard, Dictionary Learning, IEEE Signal Processing Magazine, vol.28, issue.2, pp.27-38, 2011.
DOI : 10.1109/MSP.2010.939537

R. Tibshirani, Regression shrinkage and selection via the LASSO, J. Roy. Stat. Soc. Ser. B, vol.58, issue.1, pp.267-288, 1996.
DOI : 10.1111/j.1467-9868.2011.00771.x

S. D. Babacan, R. Molina, and A. K. Katsaggelos, Bayesian Compressive Sensing Using Laplace Priors, IEEE Transactions on Image Processing, vol.19, issue.1, pp.53-63, 2010.
DOI : 10.1109/TIP.2009.2032894
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.295.319

F. Caron and A. Doucet, Sparse Bayesian nonparametric regression, Proceedings of the 25th international conference on Machine learning, ICML '08, 2008.
DOI : 10.1145/1390156.1390168
URL : https://hal.archives-ouvertes.fr/hal-00385855

M. E. Tipping, Sparse Bayesian learning and the relevance vector machine, J. Mach. Learning Research, vol.1, pp.211-244, 2001.

A. Lee, F. Caron, A. Doucet, and C. Holmes, A hierarchical Bayesian framework for constructing sparsity-inducing priors, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00518430

K. E. Themelis, A. A. Rontogiannis, and K. D. Koutroumbas, A Novel Hierarchical Bayesian Approach for Sparse Semisupervised Hyperspectral Unmixing, IEEE Transactions on Signal Processing, vol.60, issue.2, pp.585-599, 2012.
DOI : 10.1109/TSP.2011.2174052

S. Ji, Y. Xue, and L. Carin, Bayesian Compressive Sensing, IEEE Transactions on Signal Processing, vol.56, issue.6, pp.2346-2356, 2008.
DOI : 10.1109/TSP.2007.914345

D. Tzikas, A. Likas, and N. Galatsanos, Variational Bayesian Sparse Kernel-Based Blind Image Deconvolution With Student's-t Priors, IEEE Transactions on Image Processing, vol.18, issue.4, pp.753-764, 2009.
DOI : 10.1109/TIP.2008.2011757

C. Févotte and S. J. , A Bayesian Approach for Blind Separation of Sparse Sources, IEEE Transactions on Audio, Speech and Language Processing, vol.14, issue.6, pp.2174-2188, 2006.
DOI : 10.1109/TSA.2005.858523

N. Dobigeon, A. O. Hero, and J. Tourneret, Hierarchical Bayesian Sparse Image Reconstruction With Application to MRFM, IEEE Transactions on Image Processing, vol.18, issue.9, pp.2059-2070, 2009.
DOI : 10.1109/TIP.2009.2024067
URL : https://hal.archives-ouvertes.fr/hal-00324075

C. Soussen, J. Idier, D. Brie, and J. Duan, From Bernoulli–Gaussian Deconvolution to Sparse Signal Restoration, IEEE Transactions on Signal Processing, vol.59, issue.10, pp.4572-4584, 2011.
DOI : 10.1109/TSP.2011.2160633

L. Chaari, H. Batatia, N. Dobigeon, and J. Tourneret, A hierarchical sparsity-smoothness Bayesian model for 0 ? 1 ? 2 regularization, Proc. IEEE Int. Conf. Acoust., Speech, and Signal Processing (ICASSP), pp.1901-1905, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01147247

E. Remes, Sur une propriété extrémale des polynômes de Tchebychef Communications de l'Institut des Sciences Mathématiques et Mécaniques de l, pp.93-95, 1936.

T. W. Parks and J. H. Mcclellan, Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase, IEEE Transactions on Circuit Theory, vol.19, issue.2, pp.189-194, 1972.
DOI : 10.1109/TCT.1972.1083419

J. H. Mcclellan and T. W. Parks, A personal history of the Parks-McClellan algorithm, IEEE Signal Processing Magazine, vol.22, issue.2, pp.82-86, 2005.
DOI : 10.1109/MSP.2005.1406492

L. W. Neustadt, Minimum Effort Control Systems, Journal of the Society for Industrial and Applied Mathematics Series A Control, vol.1, issue.1, pp.16-31, 1962.
DOI : 10.1137/0301002

J. A. Cadzow, Algorithm for the minimum-effort problem, IEEE Transactions on Automatic Control, vol.16, issue.1, pp.60-63, 1971.
DOI : 10.1109/TAC.1971.1099634

Y. Lyubarskii and R. Vershynin, Uncertainty Principles and Vector Quantization, IEEE Transactions on Information Theory, vol.56, issue.7, pp.3491-3501, 2010.
DOI : 10.1109/TIT.2010.2048458
URL : http://arxiv.org/abs/math/0611343

Z. Cvetkovi´ccvetkovi´c, Resilience properties of redundant expansions under additive noise and quantization, IEEE Trans. Inf. Theory, vol.29, issue.3, pp.644-656, 2003.

A. R. Calderbank and I. Daubechies, The pros and cons of democracy, IEEE Transactions on Information Theory, vol.48, issue.6, pp.1721-1725, 2002.
DOI : 10.1109/TIT.2002.1003852

B. Farrell and P. Jung, A Kashin approach to the capacity of the discrete amplitude constrained Gaussian channel, Proc. Int. Conf. Sampling Theory and Applications (SAMPTA), 2009.
URL : https://hal.archives-ouvertes.fr/hal-00451439

J. Ilic and T. Strohmer, PAPR reduction in OFDM using Kashin's representation, Proc. IEEE Int. Workshop Signal Process, pp.444-448, 2009.
DOI : 10.1109/spawc.2009.5161824

C. Studer, Y. Wotao, and R. G. Baraniuk, Signal representations with minimum ?-norm, Proc. Ann. Allerton Conf. Comm. Control Comput. (Allerton), pp.1270-1277, 2012.
DOI : 10.1109/allerton.2012.6483364

C. Studer, T. Goldstein, W. Yin, and R. G. Baraniuk, Democratic representations, IEEE Trans. Inf. Theory, 2014.

J. Fuchs, Spread representations, 2011 Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR), 2011.
DOI : 10.1109/ACSSC.2011.6190120
URL : https://hal.archives-ouvertes.fr/hal-00700734

H. Jegou, T. Furon, and J. Fuchs, Anti-sparse coding for approximate nearest neighbor search, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.2012-2029
DOI : 10.1109/ICASSP.2012.6288307
URL : https://hal.archives-ouvertes.fr/hal-00661591

C. Studer and E. G. Larsson, PAR-Aware Large-Scale Multi-User MIMO-OFDM Downlink, IEEE Journal on Selected Areas in Communications, vol.31, issue.2, pp.303-313, 2013.
DOI : 10.1109/JSAC.2013.130217
URL : http://arxiv.org/abs/1202.4034

J. Tan, D. Baron, and L. Dai, Wiener Filters in Gaussian Mixture Signal Estimation With <inline-formula> <tex-math notation="TeX">\(\ell _\infty \) </tex-math></inline-formula>-Norm Error, IEEE Transactions on Information Theory, vol.60, issue.10, pp.6626-6635, 2014.
DOI : 10.1109/TIT.2014.2345260

M. Pereyra, Proximal Markov chain Monte Carlo algorithms, Statistics and Computing, vol.1, issue.4, pp.745-760, 2016.
DOI : 10.1007/s11222-015-9567-4
URL : http://arxiv.org/abs/1306.0187

C. Elvira, P. Chainais, N. Dobigeon, I. Inp-enseeiht, and U. , Bayesian anti-sparse coding ? Complementary results and supporting materials, 2016.

J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien, CR Acad. Sci. Paris Sér. A Math, vol.255, pp.2897-2899, 1962.

N. Parikh and S. Boyd, Proximal Algorithms, Foundations and Trends?? in Optimization, vol.1, issue.3, pp.127-239, 2014.
DOI : 10.1561/2400000003

L. Condat, Fast projection onto the simplex and the l1 ball, Mathematical Programming, pp.1-11, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01056171

J. Geweke, Getting It Right, Journal of the American Statistical Association, vol.99, issue.467, pp.799-804, 2004.
DOI : 10.1198/016214504000001132

N. Chopin, Fast simulation of truncated Gaussian distributions, Statistics and Computing, vol.82, issue.398, pp.275-288, 2010.
DOI : 10.1007/s11222-009-9168-1

R. Balu, T. Furon, and H. Jégou, Beyond " project and sign " for cosine estimation with binary codes, Proc. IEEE Int. Conf. Acoust., Speech, and Signal Processing (ICASSP), pp.68-884, 2014.
DOI : 10.1109/icassp.2014.6854934
URL : https://hal.archives-ouvertes.fr/hal-00942075

T. M. France and . Sc, He is currently pursuing his PhD studies at CRIStAL, under the supervision of Pierre Chainais and Nicolas Dobigeon. His research interests are centered around Bayesian nonparametrics and Markov Chain Monte Carlo (MCMC) methods with applications to signal processing, degree in applied Mathematics from the university of, 2014.

T. M. Sc, He received the Eng. degree in electrical engineering from ENSEEIHT degree in signal processing from INP Toulouse, both in 2004, the Ph.D. degree and the HabilitationàHabilitation`Habilitationà Diriger des Recherches in signal processing from INP Toulouse in 2007 and 2012, respectively. From Since 2008, he has been with INP-ENSEEIHT Toulouse, University of Toulouse, where he is currently a Professor. He conducts his research within the Signal and Communications Group, Nicolas Dobigeon (S'05?M'08?SM'13) was born His recent research activities have been focused on statistical signal and image processing Bayesian inverse problems and applications to remote sensing and biomedical imaging, 1981.