On the Convergence of Learning-based Iterative Methods for Nonconvex Inverse Problems

On the Convergence of Learning-based Iterative Methods for Nonconvex Inverse Problems

Risheng Liu 1   Shichao Cheng 1    Yi He 1    Xin Fan 1    Zhouchen Lin 2    Zhongxuan Luo 1
Dalian University of Technology 1      Peking University 2
Abstract
Numerous tasks at the core of statistics, learning and vision areas are specific cases of ill-posed inverse problems. Recently, learning-based (e.g., deep) iterative methods have been empirically shown to be useful for these problems. Nevertheless, integrating learnable structures into iterations is still a laborious process, which can only be guided by intuitions or empirical insights. Moreover, there is a lack of rigorous analysis about the convergence behaviors of these reimplemented iterations, and thus the significance of such methods is a little bit vague. This paper moves beyond these limits and proposes Flexible Iterative Modularization Algorithm (FIMA), a generic and provable paradigm for nonconvex inverse problems. Our theoretical analysis reveals that FIMA allows us to generate globally convergent trajectories for learning-based iterative methods. Meanwhile, the devised scheduling policies on flexible modules should also be beneficial for classical numerical methods in the nonconvex scenario. Extensive experiments on real applications verify the superiority of FIMA.
Algorithm
Theorem
Please refer to our paper for detailed proofs and definitions.
Convergence Analysis
Applications
Non-blind deconvolution
Blind deblur
Derain
Resource
"On the Convergence of Learning-based Iterative Methods for Nonconvex Inverse Problems"
Risheng Liu,  Shichao Cheng,  Yi He,   Xin Fan,  Zhouchen Lin,  Zhongxuan Luo

IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI) 2019