On the linear convergence rates of exchange and continuous methods for total variation minimization

Axel Flinth (1) , Frédéric de Gournay (1, 2) , Pierre Weiss (1, 2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 ITAV - Institut des Technologies Avancées en sciences du Vivant

We analyze an exchange algorithm for the numerical solution total-variation regularized inverse problems over the space M(Ω) of Radon measures on a subset Ω of R d. Our main result states that under some regularity conditions, the method eventually converges linearly. Additionally, we prove that continuously optimizing the amplitudes of positions of the target measure will succeed at a linear rate with a good initialization. Finally, we propose to combine the two approaches into an alternating method and discuss the comparative advantages of this approach.

Mots clés: inverse problems; Total variation minimization; superresolution; semi-infinite programming

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

Axel Flinth, Frédéric de Gournay, Pierre Weiss. On the linear convergence rates of exchange and continuous methods for total variation minimization. Mathematical Programming, 2020, ⟨10.1007/s10107-020-01530-0⟩. ⟨hal-02136598v2⟩