Composite Optimization by Nonconvex Majorization-Minimization
This work addresses optimization challenges in imaging, such as depth super-resolution, but is incremental as it builds on existing majorization-minimization techniques.
The paper tackles the problem of minimizing nonconvex composite functions in imaging tasks by proposing a method using nonconvex majorizers, showing it achieves globally convergent optimization and often yields superior local optima compared to previous methods when solved to global optimality.
The minimization of a nonconvex composite function can model a variety of imaging tasks. A popular class of algorithms for solving such problems are majorization-minimization techniques which iteratively approximate the composite nonconvex function by a majorizing function that is easy to minimize. Most techniques, e.g. gradient descent, utilize convex majorizers in order to guarantee that the majorizer is easy to minimize. In our work we consider a natural class of nonconvex majorizers for these functions, and show that these majorizers are still sufficient for a globally convergent optimization scheme. Numerical results illustrate that by applying this scheme, one can often obtain superior local optima compared to previous majorization-minimization methods, when the nonconvex majorizers are solved to global optimality. Finally, we illustrate the behavior of our algorithm for depth super-resolution from raw time-of-flight data.