Algorithmic Design of Majorizers for Large-Scale Inverse Problems

TL;DR

This paper introduces a duality-based method for designing diverse matrix majorizers to enhance the efficiency of majorize-minimize algorithms in large-scale inverse problems, demonstrated with 2D X-ray CT reconstruction.

Contribution

It proposes a novel duality approach for algorithmically designing matrix majorizers with various structures, overcoming computational limitations of traditional methods.

Findings

Preliminary results show accelerated convergence in 2D X-ray CT reconstruction.

The method enables the use of exotic regularizers in MM algorithms.

Potential for significant speed-ups in large-scale inverse problems.

Abstract

Iterative majorize-minimize (MM) (also called optimization transfer) algorithms solve challenging numerical optimization problems by solving a series of "easier" optimization problems that are constructed to guarantee monotonic descent of the cost function. Many MM algorithms replace a computationally expensive Hessian matrix with another more computationally convenient majorizing matrix. These majorizing matrices are often generated using various matrix inequalities, and consequently the set of available majorizers is limited to structures for which these matrix inequalities can be efficiently applied. In this paper, we present a technique to algorithmically design matrix majorizers with wide varieties of structures. We use a novel duality-based approach to avoid the high computational and memory costs of standard semidefinite programming techniques. We present some preliminary results for 2D X-ray CT reconstruction that indicate these more exotic regularizers may significantly accelerate MM algorithms.