Bayesian Pursuit Algorithms

Cedric Herzet (1) , Angélique Drémeau (2, 3)

1 FLUMINANCE - Fluid Flow Analysis, Description and Control from Image Sequences
2 IL - Institut Langevin - Ondes et Images (UMR7587)
3 Télécom ParisTech

This paper addresses the sparse representation (SR) problem within a general Bayesian framework. We show that the Lagrangian formulation of the standard SR problem, i.e. $\x^\star=\argmin_\x \lbrace \| \y-\D\x\|_2^2+\lambda\| \x\|_0 \rbrace$, can be regarded as a limit case of a general maximum a posteriori (MAP) problem involving Bernoulli-Gaussian variables. We then propose different tractable implementations of this MAP problem that we refer to as ''Bayesian pursuit algorithms". The Bayesian algorithms are shown to have strong connections with several well-known pursuit algorithms of the literature (e.g., MP, OMP, StOMP, CoSaMP, SP) and generalize them in several respects. In particular, i) they naturally allow for atom deselection; ii) they can include any prior information about the probability of occurrence of each atom within the selection process; iii) they can encompass the estimation of unkown model parameters into their recursions.

Domaines: Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Cedric Herzet, Angélique Drémeau. Bayesian Pursuit Algorithms. 2012. ⟨hal-00673801v3⟩