iPiano

iPiano: inertial proximal algorithm for nonconvex optimization. In this paper we study an algorithm for solving a minimization problem composed of a differentiable (possibly nonconvex) and a convex (possibly nondifferentiable) function. The algorithm iPiano combines forward-backward splitting with an inertial force. It can be seen as a nonsmooth split version of the Heavy-ball method from Polyak. A rigorous analysis of the algorithm for the proposed class of problems yields global convergence of the function values and the arguments. This makes the algorithm robust for usage on nonconvex problems. The convergence result is obtained based on the Kurdyka-Łojasiewicz inequality. This is a very weak restriction, which was used to prove convergence for several other gradient methods. First, an abstract convergence theorem for a generic algorithm is proved, and then iPiano is shown to satisfy the requirements of this theorem. Furthermore, a convergence rate is established for the general problem class. We demonstrate iPiano on computer vision problems – image denoising with learned priors and diffusion based image compression


References in zbMATH (referenced in 60 articles , 1 standard article )

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  1. Li, Y.; Sixou, B.; Peyrin, F.: Nonconvex mixed TV/Cahn-Hilliard functional for super-resolution/segmentation of 3D trabecular bone images (2019)
  2. Lu, Jian; Yang, Hanmei; Shen, Lixin; Zou, Yuru: Ultrasound image restoration based on a learned dictionary and a higher-order MRF (2019)
  3. Luke, D. Russell; Sabach, Shoham; Teboulle, Marc: Optimization on spheres: models and proximal algorithms with computational performance comparisons (2019)
  4. Ochs, Peter: Unifying abstract inexact convergence theorems and block coordinate variable metric iPiano (2019)
  5. Tan, Pauline; Pierre, Fabien; Nikolova, Mila: Inertial alternating generalized forward-backward splitting for image colorization (2019)
  6. Tovey, Robert; Benning, Martin; Brune, Christoph; Lagerwerf, Marinus J.; Collins, Sean M.; Leary, Rowan K.; Midgley, Paul A.; Schönlieb, Carola-Bibiane: Directional sinogram inpainting for limited angle tomography (2019)
  7. Valkonen, Tuomo: Block-proximal methods with spatially adapted acceleration (2019)
  8. Wu, Zhongming; Li, Min: General inertial proximal gradient method for a class of nonconvex nonsmooth optimization problems (2019)
  9. Bednarczuk, E. M.; Jezierska, A.; Rutkowski, K. E.: Proximal primal-dual best approximation algorithm with memory (2018)
  10. Benning, Martin; Burger, Martin: Modern regularization methods for inverse problems (2018)
  11. Boţ, Radu Ioan; Csetnek, Ernö Robert; Nimana, Nimit: An inertial proximal-gradient penalization scheme for constrained convex optimization problems (2018)
  12. Cazelles, Elsa; Seguy, Vivien; Bigot, Jérémie; Cuturi, Marco; Papadakis, Nicolas: Geodesic PCA versus log-PCA of histograms in the Wasserstein space (2018)
  13. Geiping, Jonas; Moeller, Michael: Composite optimization by nonconvex majorization-minimization (2018)
  14. Iutzeler, Franck; Malick, Jérôme: On the proximal gradient algorithm with alternated inertia (2018)
  15. Lanza, Alessandro; Morigi, Serena; Sciacchitano, Federica; Sgallari, Fiorella: Whiteness constraints in a unified variational framework for image restoration (2018)
  16. Quéau, Yvain; Durou, Jean-Denis; Aujol, Jean-François: Variational methods for normal integration (2018)
  17. Ringholm, Torbjørn; Lazić, Jasmina; Schönlieb, Carola-Bibiane: Variational image regularization with Euler’s elastica using a discrete gradient scheme (2018)
  18. Roy, Souvik; Borzì, Alfio: A new optimization approach to sparse reconstruction of log-conductivity in acousto-electric tomography (2018)
  19. Themelis, Andreas; Stella, Lorenzo; Patrinos, Panagiotis: Forward-backward envelope for the sum of two nonconvex functions: further properties and nonmonotone linesearch algorithms (2018)
  20. Wu, Chunlin; Liu, Zhifang; Wen, Shuang: A general truncated regularization framework for contrast-preserving variational signal and image restoration: motivation and implementation (2018)