  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.
