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

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.

Learning Sparse Graphs via Majorization-Minimization for Smooth Node Signals

In this letter, we propose an algorithm for learning a sparse weighted graph by estimating its adjacency matrix under the assumption that the observed signals vary smoothly over the nodes of the graph. The proposed algorithm is based on the principle of majorization-minimization (MM), wherein we first obtain a tight surrogate function for the graph learning objective and then solve the resultant surrogate problem which has a simple closed form solution. The proposed algorithm does not require tuning of any hyperparameter and it has the desirable feature of eliminating the inactive variables in the course of the iterations - which can help speeding up the algorithm. The numerical simulations conducted using both synthetic and real world (brain-network) data show that the proposed algorithm converges faster, in terms of the average number of iterations, than several existing methods in the literature.

PDMM: A novel Primal-Dual Majorization-Minimization algorithm for Poisson Phase-Retrieval problem

In this paper, we introduce a novel iterative algorithm for the problem of phase-retrieval where the measurements consist of only the magnitude of linear function of the unknown signal, and the noise in the measurements follow Poisson distribution. The proposed algorithm is based on the principle of majorization-minimization (MM); however, the application of MM here is very novel and distinct from the way MM has been usually used to solve optimization problems in the literature. More precisely, we reformulate the original minimization problem into a saddle point problem by invoking Fenchel dual representation of the log (.) term in the Poisson likelihood function. We then propose tighter surrogate functions over both primal and dual variables resulting in a double-loop MM algorithm, which we have named as Primal-Dual Majorization-Minimization (PDMM) algorithm. The iterative steps of the resulting algorithm are simple to implement and involve only computing matrix vector products. We also extend our algorithm to handle various L1 regularized Poisson phase-retrieval problems (which exploit sparsity). The proposed algorithm is compared with previously proposed algorithms such as wirtinger flow (WF), MM (conventional), and alternating direction methods of multipliers (ADMM) for the Poisson data model. The simulation results under different experimental settings show that PDMM is faster than the competing methods, and its performance in recovering the original signal is at par with the state-of-the-art algorithms.