Thresholding RULES and iterative shrinkage/thresholding algorithm: A convergence study

Matthieu Kowalski (1)

1 L2S - Laboratoire des signaux et systèmes

Imaging inverse problems can be formulated as an optimization problem and solved thanks to algorithms such as forward-backward or ISTA (Iterative Shrinkage/Thresholding Algorithm) for which non smooth functionals with sparsity constraints can be minimized efficiently. However, the soft thresholding operator involved in this algorithm leads to a biased estimation of large coefficients. That is why a step allowing to reduce this bias is introduced in practice. Indeed, in the statistical community, a large variety of thresholding operators have been studied to avoid the biased estimation of large coefficients; for instance, the non negative Garrote or the the SCAD thresholding. One can associate a non convex penalty to these opera-tors. We study the convergence properties of ISTA, possibly relaxed, with any thresholding rule and show that they correspond to a semi-convex penalty. The effectiveness of this approach is illustrated on image inverse problems.

Mots clés: relaxed ISTA; Index Terms— Sparse approximation; semi convex optimiza-tion; nonnegative garrote

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

Matthieu Kowalski. Thresholding RULES and iterative shrinkage/thresholding algorithm: A convergence study . 21st IEEE International Conference on Image Processing (ICIP 2014), Oct 2014, La Défense, Paris, France. ⟨10.1109/icip.2014.7025843⟩. ⟨hal-01102810⟩