Proximal Gradient Algorithm in the Presence of Adjoint Mismatch

Marion Savanier (1, 2) , Emilie Chouzenoux (2) , Jean-Christophe Pesquet (2) , Cyril Riddell (1) , Yves Trousset (1)

1 GE Healthcare - General Electric Medical Systems [Buc]
2 OPIS - OPtimisation Imagerie et Santé

The proximal gradient algorithm is a popular iterative algorithm to deal with penalized least-squares minimization problems. Its simplicity and versatility allow one to embed non-smooth penalties efficiently. In the context of inverse problems arising in signal and image processing, a major concern lies in the computational burden when implementing minimization algorithms. For instance, in tomographic image reconstruction, a bottleneck is the cost for applying the forward linear operator and its adjoint. Consequently, it often happens that these operators are approximated numerically, so that the adjoint property is no longer fulfilled. In this paper, we focus on the proximal gradient algorithm stability properties when such an adjoint mismatch arises. By making use of tools from convex analysis and fixed point theory, we establish conditions under which the algorithm can still converge to a fixed point. We provide bounds on the error between this point and the solution to the minimization problem. We illustrate the applicability of our theoretical results through numerical examples in the context of computed tomography.

Mots clés: Proximal gradient algorithm; adjoint mismatch; convergence analysis; fixed point methods; image reconstruction; computed tomography

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

Marion Savanier, Emilie Chouzenoux, Jean-Christophe Pesquet, Cyril Riddell, Yves Trousset. Proximal Gradient Algorithm in the Presence of Adjoint Mismatch. EUSIPCO 2020 - 28th European Signal Processing Conference, Jan 2021, Amsterdam (virtual), Netherlands. ⟨hal-02943659v2⟩