Sampling high-dimensional Gaussian distributions for general linear inverse problems

François F. Orieux (1) , Olivier Féron (2) , Jean-François Giovannelli (3)

1 IAP - Institut d'Astrophysique de Paris
2 EDF R&D OSIRIS - Optimisation, Simulation, Risque et Statistiques pour les Marchés de l’Energie
3 IMS - Laboratoire de l'intégration, du matériau au système

This paper is devoted to the problem of sampling Gaussian distributions in high dimension. Solutions exist for two specific structures of inverse covariance: sparse and circulant. The proposed algorithm is valid in a more general case especially as it emerges in linear inverse problems as well as in some hierarchical or latent Gaussian models. It relies on a perturbation-optimization principle: adequate stochastic perturbation of a criterion and optimization of the perturbed criterion. It is proved that the criterion optimizer is a sample of the target distribution. The main motivation is in inverse problems related to general (non-convolutive) linear observation models and their solution in a Bayesian framework implemented through sampling algorithms when existing samplers are infeasible. It finds a direct application in myopic,unsupervised inversion methods as well as in some non-Gaussian inversion methods. An illustration focused on hyperparameter estimation for super-resolution method shows the interest and the feasibility of the proposed algorithm.

Mots clés: Stochastic sampling; high-dimensional sampling; inverse problem; Bayesian strategy; unsupervised; myopic; semi-blind

Domaines: Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]; Calcul [stat.CO]; Applications [stat.AP]; Statistiques [math.ST]; Théorie [stat.TH]; Traitement des images [eess.IV]

François F. Orieux, Olivier Féron, Jean-François Giovannelli. Sampling high-dimensional Gaussian distributions for general linear inverse problems. IEEE Signal Processing Letters, 2012, 19 (5), pp.251. ⟨10.1109/LSP.2012.2189104⟩. ⟨hal-00779449⟩