{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T02:30:07Z","timestamp":1772332207597,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2017,12,5]],"date-time":"2017-12-05T00:00:00Z","timestamp":1512432000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this work, we propose a Bayesian online reconstruction algorithm for sparse signals based on Compressed Sensing and inspired by L1-regularization schemes. A previous work has introduced a mean-field approximation for the Bayesian online algorithm and has shown that it is possible to saturate the offline performance in the presence of Gaussian measurement noise when the signal generating distribution is known. Here, we build on these results and show that reconstruction is possible even if prior knowledge about the generation of the signal is limited, by introduction of a Laplace prior and of an extra Kullback\u2013Leibler divergence minimization step for hyper-parameter learning.<\/jats:p>","DOI":"10.3390\/e19120667","type":"journal-article","created":{"date-parts":[[2017,12,5]],"date-time":"2017-12-05T11:50:28Z","timestamp":1512474628000},"page":"667","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["L1-Minimization Algorithm for Bayesian Online Compressed Sensing"],"prefix":"10.3390","volume":"19","author":[{"given":"Paulo","family":"Rossi","sequence":"first","affiliation":[{"name":"Latam Experian DataLab , S\u00e3o Paulo-SP 04547-130, Brazil"},{"name":"Department of General Physics, Institute of Physics, University of S\u00e3o Paulo, S\u00e3o Paulo-SP 05508-090, Brazil"}]},{"given":"Renato","family":"Vicente","sequence":"additional","affiliation":[{"name":"Latam Experian DataLab , S\u00e3o Paulo-SP 04547-130, Brazil"},{"name":"Department of Applied Mathematics, Institute of Mathematics and Statistics, University of S\u00e3o Paulo, S\u00e3o Paulo-SP 05508-090, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2017,12,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1109\/LSP.2012.2224518","article-title":"Measure What Should be Measured: Progress and Challenges in Compressive Sensing","volume":"19","author":"Strohmer","year":"2012","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1109\/MSP.2007.914731","article-title":"An Introduction to Compressive Sampling","volume":"25","author":"Wakin","year":"2008","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_3","unstructured":"Holtz, O. (arXiv, 2008). Compressive sensing: A paradigm shift in signal processing, arXiv."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","article-title":"Compressed sensing","volume":"52","author":"Donoho","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Eldar, Y.C., and Kutyniok, G. (2012). Compressed Sensing: Theory and Applications, Cambridge University Press.","DOI":"10.1017\/CBO9780511794308"},{"key":"ref_6","first-page":"617","article-title":"Certain topics in telegraph transmission theory","volume":"47","author":"Nyquist","year":"1928","journal-title":"Trans. AIEE"},{"key":"ref_7","first-page":"10","article-title":"Communication in the presence of noise","volume":"37","author":"Shannon","year":"1949","journal-title":"Proc. Inst. Radio Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5406","DOI":"10.1109\/TIT.2006.885507","article-title":"Near Optimal Signal Recovery from Random Projections: Universal Encoding Strategies?","volume":"52","author":"Tao","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1109\/TIT.2005.862083","article-title":"Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information","volume":"52","author":"Romberg","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Rangan, S. (August, January 31). Generalized approximate message passing for estimation with random linear mixing. Proceedings of the 2011 IEEE International Symposium on Information Theory Proceedings (ISIT), St. Petersburg, Russia.","DOI":"10.1109\/ISIT.2011.6033942"},{"key":"ref_11","first-page":"021005","article-title":"Statistical-Physics-Based Reconstruction in Compressed Sensing","volume":"2","author":"Krzakala","year":"2012","journal-title":"Phys. Rev. X"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Tramel, E.W., Manoel, A., Caltagirone, F., Gabri\u00e9, M., and Krzakala, F. (2016, January 11\u201314). Inferring sparsity: Compressed sensing using generalized restricted Boltzmann machines. Proceedings of the 2016 IEEE Information Theory Workshop (ITW), Cambridge, UK.","DOI":"10.1109\/ITW.2016.7606837"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Krzakala, F., Mezard, M., Sausset, F., Sun, Y., and Zdeborova, L. (2012). Probabilistic reconstruction in compressed sensing: Algorithms, phase diagrams, and threshold achieving matrices. J. Stat. Mech. Theory Exp., P08009.","DOI":"10.1088\/1742-5468\/2012\/08\/P08009"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Xu, Y., Kabashima, Y., and Zdeborova, L. (2014). Bayesian signal reconstruction for 1-bit Compressed Sensing. J. Stat. Mech. Theory Exp.","DOI":"10.1088\/1742-5468\/2014\/11\/P11015"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1902","DOI":"10.1109\/TIT.2011.2177575","article-title":"Asymptotic Analysis of MAP Estimation via the Replica Method and Applications to Compressed Sensing","volume":"58","author":"Rangan","year":"2012","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Opper, M., and Winther, O. (1998). Chapter A Bayesian approach to on-line learning. On-Line Learning in Neural Networks, Cambridge University Press.","DOI":"10.1017\/CBO9780511569920.017"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1109\/TNN.2010.2046422","article-title":"Inference from aging information","volume":"21","author":"Caticha","year":"2010","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1023\/A:1007428731714","article-title":"Statistical mechanics of online learning of drifting concepts: A variational approach","volume":"32","author":"Vicente","year":"1998","journal-title":"Mach. Learn."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1115\/1.3662552","article-title":"A new approach to linear filtering and prediction problems","volume":"82","author":"Kalman","year":"1960","journal-title":"J. Basic Eng."},{"key":"ref_20","unstructured":"Stengel, R.F. (2012). Optimal Control and Estimation, Courier Corporation."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"S\u00e4rkk\u00e4, S. (2013). Bayesian Filtering and Smoothing, Cambridge University Press.","DOI":"10.1017\/CBO9781139344203"},{"key":"ref_22","unstructured":"Broderick, T., Boyd, N., Wibisono, A., Wilson, A.C., and Jordan, M.I. (2013). Streaming variational bayes. Advances in Neural Information Processing Systems, Curran."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Manoel, A., Krzakala, F., Tramel, E.W., and Zdeborov\u00e1, L. (arXiv, 2017). Streaming Bayesian inference: Theoretical limits and mini-batch approximate message-passing, arXiv.","DOI":"10.1109\/ALLERTON.2017.8262853"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"022137","DOI":"10.1103\/PhysRevE.94.022137","article-title":"Bayesian online compressed sensing","volume":"94","author":"Rossi","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","article-title":"Regression Shrinkage and Selection via the Lasso","volume":"58","author":"Tibshirani","year":"1996","journal-title":"J. R. Stat. Soc. Ser. B (Methodological)"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"188701","DOI":"10.1103\/PhysRevLett.104.188701","article-title":"Statistical Mechanics of Compressed Sensing","volume":"104","author":"Ganguli","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"L09003","DOI":"10.1088\/1742-5468\/2009\/09\/L09003","article-title":"A typical reconstruction limit for compressed sensing based on Lp-norm minimization","volume":"9","author":"Kabashima","year":"2009","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1109\/TSP.2009.2027773","article-title":"Bayesian Compressive Sensing Via Belief Propagation","volume":"58","author":"Baron","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"835","DOI":"10.1093\/biomet\/asp047","article-title":"Bayesian lasso regression","volume":"96","author":"Hans","year":"2009","journal-title":"Biometrika"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1198\/016214508000000337","article-title":"The bayesian lasso","volume":"103","author":"Park","year":"2008","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1150","DOI":"10.1109\/TPAMI.2003.1227989","article-title":"Adaptive sparseness for supervised learning","volume":"25","author":"Figueiredo","year":"2003","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1214\/009053604000000067","article-title":"Least angle regression","volume":"32","author":"Efron","year":"2004","journal-title":"Ann. Stat."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Barber, D. (2012). Bayesian Reasoning and Machine Learning, Cambridge University Press.","DOI":"10.1017\/CBO9780511804779"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/12\/667\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:52:41Z","timestamp":1760208761000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/12\/667"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,5]]},"references-count":33,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2017,12]]}},"alternative-id":["e19120667"],"URL":"https:\/\/doi.org\/10.3390\/e19120667","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,12,5]]}}}