Minorization-maximization Research Articles

A new MM algorithm for root‐finding problems

The minorization–maximization (MM) algorithm is an optimization technique for iteratively calculating the maximizer of a concave target function rather than a root–finding tool. In this paper, we in the first time develop the MM algorithm as a new method for seeking the root of a univariate nonlinear equation . The key idea is to transfer the root–finding issue to iteratively calculate the maximizer of a concave target function by designing a new MM algorithm. According to the ascent property of the MM algorithm, we know that the proposed algorithm converges to the root and does not depend on any initial values, in contrast to Newton's method. Several statistical examples are provided to demonstrate the proposed algorithm.

Modeling intra-individual inter-trial EEG response variability in autism.

Autism spectrum disorder (autism) is a prevalent neurodevelopmental condition characterized by early emerging impairments in social behavior and communication. EEG represents a powerful and non-invasive tool for examining functional brain differences in autism. Recent EEG evidence suggests that greater intra-individual trial-to-trial variability across EEG responses in stimulus-related tasks may characterize brain differences in autism. Traditional analysis of EEG data largely focuses on mean trends of the trial-averaged data, where trial-level analysis is rarely performed due to low neural signal to noise ratio. We propose to use nonlinear (shape-invariant) mixed effects (NLME) models to study intra-individual inter-trial EEG response variability using trial-level EEG data. By providing more precise metrics of response variability, this approach could enrich our understanding of neural disparities in autism and potentially aid the identification of objective markers. The proposed multilevel NLME models quantify variability in the signal's interpretable and widely recognized features (e.g., latency and amplitude) while also regularizing estimation based on noisy trial-level data. Even though NLME models have been studied for more than three decades, existing methods cannot scale up to large data sets. We propose computationally feasible estimation and inference methods via the use of a novel minorization-maximization (MM) algorithm. Extensive simulations are conducted to show the efficacy of the proposed procedures. Applications to data from a large national consortium find that children with autism have larger intra-individual inter-trial variability in P1 latency in a visual evoked potential (VEP) task, compared to their neurotypical peers.

Statistical spatial information based power optimization algorithm for multi-user massive MIMO systems

Power optimization has a significant role in multi-user massive multiple-input and multiple-output (mMIMO) wireless communication systems, whereas the acquisition of instantaneous channel state information (I-CSI) is a serious challenge in practical scenarios. The statistical spatial information (SPI), which contains angle-of-departures (AoDs) and corresponding large-scale fading factors, is slowly varying. The power optimization based on SPI to improve system performance needs to be explored for multi-user mMIMO systems. In this paper, two novel algorithms for SPI-based power optimization are proposed to maximize the ergodic sum-rate (ESR) under total power constraint for the system. Specifically, the minorization-maximization-based power optimization (MM-PO) algorithm and the closed-form power optimization (CF-PO) algorithm are proposed utilizing the minorization-maximization (MM) method and the fractional programming (FP) method, respectively. Moreover, the ESR is expressed based on the extracted SPI. The approximate expression of the ESR is derived. The complexity of the algorithms depends on the number of users, and the algorithms are scalable in the system. The proposed algorithms can effectively achieve the advantages of SPI while being robust to the uncertainty of I-CSI. Simulation results verify the ESR of our two proposed algorithms is 1.47 and 1.4 times higher than that of the equal power allocation algorithm at a high SNR when the number of antenna and user are equal to 64 and 20, respectively.

A hierarchical count data clustering based on Multinomial Nested Dirichlet Mixture using the Minorization-Maximization framework

Despite the widespread acceptance of mixture models among researchers, obtaining good results involves several challenges. In this paper, we address two main challenges: parameter estimation and data representation strategies.Expectation-Maximization (EM) is a widely used framework for parameter estimation. However, many factors complicate the process through intractable calculations of the posterior distribution and the parameters. Minorization-Maximization (MM) is an alternative framework that relaxes the complications and requirements of the EM. This paper adopts the MM framework for the Multinomial Nested Dirichlet Mixture in a hierarchical manner. The hierarchical nature of the MNDM is exploited through a Hierarchical Feature Learning framework (HFL), where the data represented is a result of the well-known Spatial Pyramid Matching method. Moreover, the mixture's components are determined by the Minimum Message Length (MML). Therefore, this paper presents an HFL framework for the data representation of the MNDM, where its learning is based on the MM framework, and its selection is based on MML. The validation of the two addressed improvements is proven through three visual datasets using recall and precision performance metrics.

RIS-aided double beamforming optimization algorithm for improving secrecy rate in space–ground integrated networks

The high-frequency band, crucial for supporting the 5G/6G system, faces challenges of signal obstruction by obstacles. This is attributed to significant path loss resulting from radio straightness and short radio distance. To address these challenges, there is a growing interest in leveraging non-terrestrial networks (NTNs) and reconfigurable intelligent surfaces (RISs), utilizing high-altitude satellites as base stations or terminals. Within a three-dimensional NTN system, the vulnerability of wireless signals to eavesdropping due to the open nature of the environment is a notable drawback. To mitigate this vulnerability, this paper introduces an algorithm designed to maximize the secrecy rate. The proposed algorithm optimizes security performance by fine-tuning the base station and RIS beamforming vectors. This optimization is achieved through successive convex approximation and minorization–maximization algorithms. Simulation results affirm the superiority of the proposed algorithm in terms of secrecy rate over existing techniques.

Robust Transmission Design for RIS-Assisted Secure Multiuser Communication Systems in the Presence of Hardware Impairments

This paper investigates reconfigurable intelligent surface (RIS)-assisted secure multiuser communication systems in the presence of hardware impairments (HIs) at the RIS and the transceivers. We jointly optimize the beamforming vectors at the base station (BS) and the phase shifts of the reflecting elements at the RIS so as to maximize the weighted minimum approximate ergodic secrecy rate (WMAESR), subject to the transmission power constraints at the BS and unit-modulus constraints at the RIS. To solve the formulated optimization problem, we first decouple it into two tractable subproblems and then use the block coordinate descent (BCD) method to alternately optimize the subproblems. Two different methods are proposed to solve the two obtained subproblems. The first method transforms each subproblem into a second order cone programming (SOCP) problem by invoking the penalty convexconcave procedure (CCP) method and the closed-form fractional programming (FP) criterion, and then directly solves them by using CVX. The second method leverages the minorization-maximization (MM) algorithm. Specifically, we first derive a concave approximation function, which is a lower bound of the original objective function, and then the two subproblems are transformed into two simple surrogate problems that admit closed-form solutions. Simulation results verify the performance gains of the proposed robust transmission methods over existing non-robust designs. In addition, the MM algorithm is shown to have much lower complexity than the SOCP-based algorithm.

Smart Grid Outlier Detection Based on the Minorization-Maximization Algorithm.

Outliers can be generated in the power system due to aging system equipment, faulty sensors, incorrect line connections, etc. The existence of these outliers will pose a threat to the safe operation of the power system, reduce the quality of the data, affect the completeness and accuracy of the data, and thus affect the monitoring analysis and control of the power system. Therefore, timely identification and treatment of outliers are essential to ensure stable and reliable operation of the power system. In this paper, we consider the problem of detecting and localizing outliers in power systems. The paper proposes a Minorization-Maximization (MM) algorithm for outlier detection and localization and an estimation of unknown parameters of the Gaussian mixture model (GMM). To verify the performance of the method, we conduct simulation experiments by simulating different test scenarios in the IEEE 14-bus system. Numerical examples show that in the presence of outliers, the MM algorithm can detect outliers better than the traditional algorithm and can accurately locate outliers with a probability of more than 95%. Therefore, the algorithm provides an effective method for the handling of outliers in the power system, which helps to improve the monitoring analyzing and controlling ability of the power system and to ensure the stable and reliable operation of the power system.

Hardware-Impaired Rician-Faded Cell-Free Massive MIMO Systems With LSFD and Channel Aging: SE Analysis and Optimization

We study the impact of channel aging on the uplink of a cell-free (CF) massive multiple-input multiple-output (mMIMO) system by considering i) spatially-correlated Rician-faded channels; ii) hardware impairments at the access points and user equipments (UEs); and iii) two-layer large-scale fading decoding (LSFD). We first derive a closed-form spectral efficiency (SE) expression for this system, and later propose two novel optimization techniques to optimize the non-convex SE metric by exploiting the minorization-maximization (MM) method. The first one requires a numerical optimization solver, and has a high computation complexity. The second one with closed-form transmit power updates, has a trivial computation complexity. We numerically show that i) the two-layer LSFD scheme effectively mitigates the interference due to channel aging for both low- and high-velocity UEs; and ii) increasing the number of AP antennas does not mitigate the SE deterioration due to channel aging. We numerically characterize the optimal pilot length required to maximize the SE for various UE speeds. We also numerically show that the proposed closed-form MM optimization yields the same SE as that of the first technique, which requires numerical solver, and that too with a much reduced time-complexity.

Analysis and Optimization of Hardware Impaired FD Rician-Faded mMIMO Systems

We consider a full duplex (FD) multi-cell massive multi-input-multi-output (mMIMO) network, wherein FD mMIMO base-stations (BSs) serve multiple FD user equipments (UEs) on the same time-frequency resource. We derive closed-form uplink and downlink spectral efficiency (SE) expressions by considering i) radio-frequency impairments at the BSs and the UEs; ii) dynamic-resolution analog-to-digital converter/digital-to-analog converter (ADC/DAC) architecture at the BSs and low-resolution ADC/DAC at the UEs; and iii) spatially-correlated Rician channels. We optimize the non-convex global energy efficiency metric for this FD system by using the minorization-maximization (MM) framework, which transforms the optimization into a sequence of surrogate concave problems. We construct a novel surrogate function to exploit the MM framework. We numerically characterize the effect of FD interference on Rayleigh- and Rician-faded mMIMO networks, and show the higher immunity of the latter to the FD interference. We also show the SE gain provided by the dynamic ADC/DAC architecture which allows us to judiciously choose ADC/DAC resolution, while reducing the power consumption.

Resource Allocation for IRS-Assisted Uplink URLLC Systems

We study the resource allocation optimization in an intelligent reflecting surface (IRS)-assisted uplink ultra reliable and low-latency communication (URLLC) system. The transmit power of users, the receive beamformer at the base station (BS) and the IRS reflection phase shifts are jointly optimized to maximize the achievable sum finite blocklength (FBL) rate, which turns out to be a complicated non-convex problem. To tackle this problem, we develop a low-complexity minorization-maximization (MM)-based algorithm, which only involves calculating closed-form expressions in each iteration. Compared with an existing quadratic transform based scheme, the proposed design achieves better performance while with reduced complexity.

A Fast and Efficient Estimation of the Parameters of a Model of Accident Frequencies via an MM Algorithm

In this paper, we consider a multivariate statistical model of accident frequencies having a variable number of parameters and whose parameters are dependent and subject to box constraints and linear equality constraints. We design a minorization-maximization (MM) algorithm and an accelerated MM algorithm to compute the maximum likelihood estimates of the parameters. We illustrate, through simulations, the performance of our proposed MM algorithm and its accelerated version by comparing them to Newton-Raphson (NR) and quasi-Newton algorithms. The results suggest that the MM algorithm and its accelerated version are better in terms of convergence proportion and, as the number of parameters increases, they are also better in terms of computation time.

Energy Minimization in RIS-Assisted UAV-Enabled Wireless Power Transfer Systems

Unmanned aerial vehicle (UAV)-enabled wireless power transfer (WPT) systems offer significant advantages in coverage and deployment flexibility, but suffer from endurance limitations due to the limited onboard energy. This paper proposes to improve the energy efficiency of UAV-enabled WPT systems with multiple ground sensors by utilizing reconfigurable intelligent surface (RIS). Specifically, the total energy consumption of the UAV is minimized, while meeting the energy requirement of each sensor. Firstly, we consider a fly-hover-broadcast (FHB) protocol, in which the UAV radiates radio frequency (RF) signals only at several hovering locations. The energy minimization problem is formulated to jointly optimize the UAV’s trajectory, hovering time and the RIS’s reflection coefficients. To solve this complex non-convex problem, we propose an efficient algorithm. Specifically, the successive convex approximation (SCA) framework is adopted to jointly optimize the UAV’s trajectory and hovering time, in which a minorization-maximization (MM) algorithm that maximizes the minimum charged energy of all sensors is provided to update the reflection coefficients. Then, we investigate the general scenario in which the RF signals are radiated during the flight, aiming to minimize the total energy consumption of the UAV by jointly optimizing the UAV’s trajectory, flight time and the RIS’s reflection coefficients. By applying the path discretization (PD) protocol, the optimization problem is formulated with a finite number of variables. A high-quality solution for this more challenging problem is obtained. Finally, our simulation results demonstrate the effectiveness of the proposed algorithm and the benefits of RIS in energy saving.

Adaptive Radar Detection and Bearing Estimation in the Presence of Unknown Mutual Coupling

This paper deals with joint adaptive radar detection and target bearing estimation in the presence of mutual coupling among the array elements. First of all, a suitable model of the signal received by the multichannel radar is developed via a linearization procedure of the Uniform Linear Array (ULA) manifold around the nominal array looking direction together with the use of symmetric Toeplitz structured matrices to represent the mutual coupling effects. Hence, the Generalized Likelihood Ratio Test (GLRT) detector is evaluated under the assumption of homogeneous radar environment. Its computation leverages a specific Minorization-Maximization (MM) framework, with proven convergence properties, to optimize the concentrated likelihood function under the target presence hypothesis. Besides, when the number of active mutual coupling coefficients is unknown, a Multifamily Likelihood Ratio Test (MFLRT) approach is invoked. During the analysis phase, the performance of the new detectors is compared with benchmarks as well as with counterparts available in the open literature which neglect the mutual coupling phenomenon. The results indicate that it is necessary to consider judiciously the coupling effect since the design phase, to guarantee performance levels close to the benchmark.

Multi-Spectrally Constrained Low-PAPR Waveform Optimization for MIMO Radar Space-Time Adaptive Processing

This paper focuses on the joint design of transmit waveforms and receive filters for airborne multiple-input-multiple-output (MIMO) radar systems in spectrally crowded environments. The purpose is to maximize the output signal-to-interference-plus-noise-ratio (SINR) in the presence of signal-dependent clutter. To improve the practicability of the radar waveforms, both a multi-spectral constraint and a peak-to-average-power ratio (PAPR) constraint are imposed. A cyclic method is derived to iteratively optimize the transmit waveforms and receive filters. In particular, to tackle the encountered non-convex constrained fractional programming in designing the waveforms (for fixed filters), we resort to the Dinkelbach's transform, minorization-maximization (MM), and leverage the alternating direction method of multipliers (ADMM). We highlight that the proposed algorithm can iterate from an infeasible initial point and the waveforms at convergence not only satisfy the stringent constraints, but also attain superior performance.

Efficient algorithms for survival data with multiple outcomes using the frailty model.

Survival data with multiple outcomes are frequently encountered in biomedical investigations. An illustrative example comes from Alzheimer's Disease Neuroimaging Initiative study where the cognitively normal subjects may clinically progress to mild cognitive impairment and/or Alzheimer's disease dementia. Transition time from normal cognition to mild cognitive impairment and that from mild cognitive impairment to Alzheimer's disease are expected to be correlated within subjects and the dependence is often accommodated by the frailty (random effects). Estimation in the frailty model unavoidably involves multiple integrations which may be intractable and hence leads to severe computational challenges, especially in the presence of high-dimensional covariates. In this paper, we propose efficient minorization-maximization algorithms in the frailty model for survival data with multiple outcomes. The alternating direction method of multipliers is further incorporated for simultaneous variable selection and homogeneity pursuit via regularization and fusion. Extensive simulation studies are conducted to assess the performance of the proposed algorithms. An application to the Alzheimer's Disease Neuroimaging Initiative data is also provided to illustrate their practical utilities.

An MM Algorithm for the Frailty-Based Illness Death Model with Semi-Competing Risks Data

For analyzing multiple events data, the illness death model is often used to investigate the covariate–response association for its easy and direct interpretation as well as the flexibility to accommodate the within-subject dependence. The resulting estimation and inferential procedures often depend on the subjective specification of the parametric frailty distribution. For certain frailty distributions, the computation can be challenging as the estimation involves both the nonparametric component and the parametric component. In this paper, we develop efficient computational methods for analyzing semi-competing risks data in the illness death model with the general frailty, where the Minorization–Maximization (MM) principle is employed for yielding accurate estimation and inferential procedures. Simulation studies are conducted to assess the finite-sample performance of the proposed method. An application to a real data is also provided for illustration.

Radar and Communication Spectral Coexistence on Moving Platform with Interference Suppression

With the development of intelligent transportation, radar and communication on moving platforms are competing for the spectrum. In this paper, we propose and demonstrate a new algorithmic framework for radar-communication spectral coexistence system on moving platform with mutual interference suppression, in which communication rate and the radar signal-to-interference-plus-noise ratio (SINR) are simultaneously optimized, under the energy constraints for the two systems and the radar constant modulus constraint. The radar spatial-temporal filter at the receiver and transmitting waveform are optimized, while the codebook matrix is optimized for the communication system. To cope with the established non-convex problem with triplet variables, we decouple the original problem into multiple subproblems, for which an alternating algorithm based on iterative procedures is derived with lower computational complexity. Specifically, the subproblems of communication codebook and radar filter design are convex and the closed-form solutions can be easily obtained, while the radar waveform optimization is non-convex. Then we propose a novel scheme by exploiting the alternating direction method of multipliers (ADMM) based on minorization-maximization (MM) framework. Finally, to reveal the effectiveness of the proposed algorithm in different scenarios, numerical results are provided.

Efficient Estimation and Inference in the Proportional Odds Model for Survival Data

In modeling time-to-event data with long-term survivors, the proportional hazards model is widely used for its easy and direct interpretation as well as the flexibility to accommodate the past information and allow time-varying predictors. This becomes most relevant when the mortality of individuals converges with time, and the estimation and inference based upon the proportional odds model can often yield more accurate and reasonable results than the classical Cox’s proportional hazards model. Along with the fast development of the data science technologies, computational challenges for survival data with increasing sample size and diverging parameter dimension exist. Currently, existing methods for analyzing such data are computationally inconvenient. In this paper, we propose efficient computational methods for analyzing survival data in the proportional odds model, where the nonparametric maximum likelihood approach is combined with the minorization-maximization (MM) algorithm and the regularization scheme to yield fast and accurate estimation and inferential procedures. The illustration of the methodology using extensive simulation studies and then the application to the Veterans’ Administration lung cancer data is also given.

Joint Beamforming and Trajectory Design for Aerial Intelligent Reflecting Surface-Aided Secure Transmission

This paper studies the secure transmission challenge confronted with future communication systems. In our considered model, the confidential communication between legitimate users is strengthened by an aerial intelligent reflecting surface (AIRS) deployed on aerial platforms such as an unmanned aerial vehicle (UAV). The average secrecy rate for all time slots is first investigated to improve information security during AIRS flights. Then, the transmit beamforming, phase-shifting matrix, and trajectory of AIRS are jointly designed, aiming to maximize the average secrecy rate performance between legitimate users. On account of the non-convexity of the formulated objective function and the coupled three key variables, we resort to an alternative strategy that converts the original objective into three sub-problems and solves them recursively. In particular, the transmit beamforming is designed on the basis of the generalized eigenvalue optimization method, and the closed-form solution is derived. For the AIRS-related optimization, a Minorization-Maximization (MM)-based algorithm and a deep deterministic policy gradient (DDPG)-based method are proposed to derive the solutions of phase shift matrix and trajectory, respectively. Simulation results show that AIRS assistance can obtain nearly twice the secrecy performance of terrestrial intelligent reflecting surface (TIRS) systems.

Cooperative Design for MIMO Radar- Communication Spectral Sharing System Based on Mutual Information Optimization

This paper considers a novel cooperative design for radar-communication spectral sharing system with multiple-input multiple-output (MIMO) structure based on mutual information optimization. We present a new spectral sharing framework aiming at maximizing the radar mutual information. Specifically, the transmit sequences including radar waveform and communication codebook are designed simultaneously based on radar and communication system mutual information optimization. Subsequently, an iterative procedure is devised to maximize the radar mutual information under the constraints including the transmit power, the constant modulus requirement for radar waveform, and the communication system mutual information performance requirement constraint. The proposed approach exploits the alternating optimization based on the Minorization-Maximization (MM) framework, Lagrange algorithm, and the alternating direction method of multipliers (ADMM) techniques to decompose the original design into two simpler subproblems with a closed-form solution. Simulation results are presented to demonstrate the advantage of the proposed schemes.