Composite Optimization by Nonconvex Majorization-Minimization

TL;DR

This paper introduces a globally convergent optimization scheme for nonconvex composite functions using nonconvex majorizers, demonstrating improved local optima in imaging tasks like depth super-resolution.

Contribution

It proposes a novel class of nonconvex majorizers for composite functions, ensuring global convergence and superior local optima in nonconvex optimization.

Findings

Numerical results show improved local optima with the new scheme.

The method guarantees global convergence for a class of nonconvex majorizers.

Application to depth super-resolution demonstrates practical effectiveness.

Abstract

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.