An Oracle Inequality for Quasi-Bayesian Non-Negative Matrix Factorization

The aim of this paper is to provide some theoretical understanding of quasi-Bayesian aggregation methods non-negative matrix factorization. We derive an oracle inequality for an aggregated estimator. This result holds for a very general class of prior distributions and shows how the prior affects the rate of convergence.Comment: This is the corrected version of the published paper P. Alquier, B. Guedj, An Oracle Inequality for Quasi-Bayesian Non-negative Matrix Factorization, Mathematical Methods of Statistics, 2017, vol. 26, no. 1, pp. 55-67. Since then Arnak Dalalyan (ENSAE) found a mistake in the proofs. We fixed the mistake at the price of a slightly different logarithmic term in the boun

Reversible jump MCMC for non-negative matrix factorization

We present a fully Bayesian approach to Non- Negative Matrix Factorisation (NMF) by developing a Reversible Jump Markov Chain Monte Carlo (RJMCMC) method which provides full posteriors over the matrix components. In addition the NMF model selection issue is addressed, for the first time, as our RJMCMC procedure provides the posterior distribution over the matrix dimensions and therefore the number of components in the NMF model. A comparative analysis is provided with the Bayesian Information Criterion (BIC) and model selection employing estimates of the marginal likelihood. An illustrative synthetic example is provided using blind mixtures of images. This is then followed by a large scale study of the recovery of component spectra from multiplexed Raman readouts. The power and flexibility of the Bayesian methodology and the proposed RJMCMC procedure to objectively assess differing model structures and infer the corresponding plausible component spectra for this complex data is demonstrated convincingly. © 2009 by the authors

Inférence bayésienne dans des problèmes inverses, myopes et aveugles en traitement du signal et des images

Les activités de recherche présentées concernent la résolution de problèmes inverses, myopes et aveugles rencontrés en traitement du signal et des images. Les méthodes de résolution privilégiées reposent sur une démarche d'inférence bayésienne. Celle-ci offre un cadre d'étude générique pour régulariser les problèmes généralement mal posés en exploitant les contraintes inhérentes aux modèles d'observation. L'estimation des paramètres d'intérêt est menée à l'aide d'algorithmes de Monte Carlo qui permettent d'explorer l'espace des solutions admissibles. Un des domaines d'application visé par ces travaux est l'imagerie hyperspectrale et, plus spécifiquement, le démélange spectral. Le second travail présenté concerne la reconstruction d'images parcimonieuses acquises par un microscope MRFM

Reversible Jump MCMC for Non-Negative matrix factorization with application to Raman spectral decomposition

We present a fully Bayesian approach to Non-Negative Matrix Factorisation (NMF) by developing a Reversible Jump Markov Chain Monte Carlo (RJMCMC) method which provides full posteriors over the matrix components. In addition the NMF model selection issue is addressed, for the first time, as our RJMCMC procedure provides the posterior distribution over the matrix dimensions and therefore the number of components in the NMF model. A comparative analysis is provided with the Bayesian Information Criterion (BIC) and model selection employing estimates of the marginal likelihood. An illustrative synthetic example is provided using blind mixtures of images. This is then followed by a large scale study of the recovery of component spectra from multiplexed Raman readouts. The power and flexibility of the Bayesian methodology and the proposed RJMCMC procedure to objectively assess differing model structures and infer the corresponding plausible component spectra for this complex data is demonstrated convincingly