Min-Max Framework for Majorization-Minimization Algorithms in Signal Processing Applications: An Overview

TL;DR

This paper introduces the Min-Max Framework for Majorization-Minimization (MM4MM), a versatile approach for solving complex non-convex optimization problems in signal processing by reformulating them as min-max problems.

Contribution

It provides a comprehensive theoretical foundation and practical guidelines for developing MM4MM algorithms applicable to various signal processing tasks, including detailed examples and applications.

Findings

MM4MM algorithms are hyper-parameter free.

The approach guarantees monotonic convergence.

Successful application in ten signal processing problems.

Abstract

This monograph presents a theoretical background and a broad introduction to the Min-Max Framework for Majorization-Minimization (MM4MM), an algorithmic methodology for solving minimization problems by formulating them as min-max problems and then employing majorization-minimization. The monograph lays out the mathematical basis of the approach used to reformulate a minimization problem as a min-max problem. With the prerequisites covered, including multiple illustrations of the formulations for convex and non-convex functions, this work serves as a guide for developing MM4MM-based algorithms for solving non-convex optimization problems in various areas of signal processing. As special cases, we discuss using the majorization-minimization technique to solve min-max problems encountered in signal processing applications and min-max problems formulated using the Lagrangian. Lastly, we present detailed examples of using MM4MM in ten signal processing applications such as phase retrieval, source localization, independent vector analysis, beamforming and optimal sensor placement in wireless sensor networks. The devised MM4MM algorithms are free of hyper-parameters and enjoy the advantages inherited from the use of the majorization-minimization technique such as monotonicity.