arXiv subjects

Prabhu Babu

Publications and source records attributed to Prabhu Babu.

At least 19 recordsLinked to original sources

arXiv2025

Majorization-Minimization based Hybrid Localization Method for High Precision Localization in Wireless Sensor Networks

This paper investigates the hybrid source localization problem using the four radio measurements - time of arrival (TOA), time difference of arrival (TDOA), received signal strength (RSS), and angle of arrival (AOA). First, after invoking tractable approximations in the RSS and AOA models, the maximum likelihood estimation (MLE) problem for the hybrid TOA-TDOA-RSS-AOA data model is derived. Then a weighted least-squares problem is formulated from the MLE, which is solved using the principle of the majorization-minimization (MM), resulting in an iterative algorithm with guaranteed convergence. The key feature of the proposed method is that it provides a unified framework where localization using any possible merger out of these four measurements can be implemented as per the requirement/application. Extensive numerical simulations are conducted to study the performance of the proposed method. The obtained results indicate that the hybrid localization model improves the localization accuracy compared to the heterogeneous measurements under different network scenarios, which also includes the presence of non-line of sight (NLOS) errors.

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arXiv2025

Dual-Function Beamforming Design For Multi-Target Localization and Reliable Communications

This paper investigates the transmit beamforming design for multiple-input multiple-output systems to support both multi-target localization and multi-user communications. To enhance the target localization performance, we derive the asymptotic Cramér-Rao bound (CRB) for target angle estimation by assuming that the receive array is linear and uniform. Then we formulate a beamforming design problem based on minimizing an upper bound on the asymptotic CRB (which is shown to be equivalent to {maximizing} the harmonic mean of the weighted beampattern responses at the target directions). Moreover, we impose a constraint on the SINR of each received communication signal to guarantee reliable communication performance. Two iterative algorithms are derived to tackle the non-convex design problem: one is based on the alternating direction method of multipliers, and the other uses the majorization-minimization technique to solve an equivalent minimax problem. Numerical results show that, through elaborate dual-function beamforming matrix design, the proposed algorithms can simultaneously achieve superior angle estimation performance as well as high-quality multi-user communications.

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arXiv2024

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.

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arXiv2023

New Methods for MLE of Toeplitz Structured Covariance Matrices with Applications to RADAR Problems

This work considers Maximum Likelihood Estimation (MLE) of a Toeplitz structured covariance matrix. In this regard, an equivalent reformulation of the MLE problem is introduced and two iterative algorithms are proposed for the optimization of the equivalent statistical learning framework. Both the strategies are based on the Majorization Minimization (MM) paradigm and hence enjoy nice properties such as monotonicity and ensured convergence to a stationary point of the equivalent MLE problem. The proposed framework is also extended to deal with MLE of other practically relevant covariance structures, namely, the banded Toeplitz, block Toeplitz, and Toeplitz-block-Toeplitz. Through numerical simulations, it is shown that the new methods provide excellent performance levels in terms of both mean square estimation error (which is very close to the benchmark Cramér-Rao Bound (CRB)) and signal-to-interference-plus-noise ratio, especially in comparison with state of the art strategies.

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arXiv2023

Two new algorithms for maximum likelihood estimation of sparse covariance matrices with applications to graphical modeling

In this paper, we propose two new algorithms for maximum-likelihood estimation (MLE) of high dimensional sparse covariance matrices. Unlike most of the state of-the-art methods, which either use regularization techniques or penalize the likelihood to impose sparsity, we solve the MLE problem based on an estimated covariance graph. More specifically, we propose a two-stage procedure: in the first stage, we determine the sparsity pattern of the target covariance matrix (in other words the marginal independence in the covariance graph under a Gaussian graphical model) using the multiple hypothesis testing method of false discovery rate (FDR), and in the second stage we use either a block coordinate descent approach to estimate the non-zero values or a proximal distance approach that penalizes the distance between the estimated covariance graph and the target covariance matrix. Doing so gives rise to two different methods, each with its own advantage: the coordinate descent approach does not require tuning of any hyper-parameters, whereas the proximal distance approach is computationally fast but requires a careful tuning of the penalty parameter. Both methods are effective even in cases where the number of observed samples is less than the dimension of the data. For performance evaluation, we test the proposed methods on both simulated and real-world data and show that they provide more accurate estimates of the sparse covariance matrix than two state-of-the-art methods.

stat.ME↗

arXiv2023

Fair principal component analysis (PCA): minorization-maximization algorithms for Fair PCA, Fair Robust PCA and Fair Sparse PCA

In this paper we propose a new iterative algorithm to solve the fair PCA (FPCA) problem. We start with the max-min fair PCA formulation originally proposed in [1] and derive a simple and efficient iterative algorithm which is based on the minorization-maximization (MM) approach. The proposed algorithm relies on the relaxation of a semi-orthogonality constraint which is proved to be tight at every iteration of the algorithm. The vanilla version of the proposed algorithm requires solving a semi-definite program (SDP) at every iteration, which can be further simplified to a quadratic program by formulating the dual of the surrogate maximization problem. We also propose two important reformulations of the fair PCA problem: a) fair robust PCA -- which can handle outliers in the data, and b) fair sparse PCA -- which can enforce sparsity on the estimated fair principal components. The proposed algorithms are computationally efficient and monotonically increase their respective design objectives at every iteration. An added feature of the proposed algorithms is that they do not require the selection of any hyperparameter (except for the fair sparse PCA case where a penalty parameter that controls the sparsity has to be chosen by the user). We numerically compare the performance of the proposed methods with two of the state-of-the-art approaches on synthetic data sets and a real-life data set.

stat.ML↗

arXiv2023

Pearson-Matthews correlation coefficients for binary and multinary classification and hypothesis testing

The Pearson-Matthews correlation coefficient (usually abbreviated MCC) is considered to be one of the most useful metrics for the performance of a binary classification or hypothesis testing method (for the sake of conciseness we will use the classification terminology throughout, but the concepts and methods discussed in the paper apply verbatim to hypothesis testing as well). For multinary classification tasks (with more than two classes) the existing extension of MCC, commonly called the $\text{R}_{\text{K}}$ metric, has also been successfully used in many applications. The present paper begins with an introductory discussion on certain aspects of MCC. Then we go on to discuss the topic of multinary classification that is the main focus of this paper and which, despite its practical and theoretical importance, appears to be less developed than the topic of binary classification. Our discussion of the $\text{R}_{\text{K}}$ is followed by the introduction of two other metrics for multinary classification derived from the multivariate Pearson correlation (MPC) coefficients. We show that both $\text{R}_{\text{K}}$ and the MPC metrics suffer from the problem of not decisively indicating poor classification results when they should, and introduce three new enhanced metrics that do not suffer from this problem. We also present an additional new metric for multinary classification which can be viewed as a direct extension of MCC.

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arXiv2023

Low-rank covariance matrix estimation for factor analysis in anisotropic noise: application to array processing and portfolio selection

Factor analysis (FA) or principal component analysis (PCA) models the covariance matrix of the observed data as R = SS' + Σ, where SS' is the low-rank covariance matrix of the factors (aka latent variables) and Σ is the diagonal matrix of the noise. When the noise is anisotropic (aka nonuniform in the signal processing literature and heteroscedastic in the statistical literature), the diagonal elements of Σ cannot be assumed to be identical and they must be estimated jointly with the elements of SS'. The problem of estimating SS' and Σ in the above covariance model is the central theme of the present paper. After stating this problem in a more formal way, we review the main existing algorithms for solving it. We then go on to show that these algorithms have reliability issues (such as lack of convergence or convergence to infeasible solutions) and therefore they may not be the best possible choice for practical applications. Next we explain how to modify one of these algorithms to improve its convergence properties and we also introduce a new method that we call FAAN (Factor Analysis for Anisotropic Noise). FAAN is a coordinate descent algorithm that iteratively maximizes the normal likelihood function, which is easy to implement in a numerically efficient manner and has excellent convergence properties as illustrated by the numerical examples presented in the paper. Out of the many possible applications of FAAN we focus on the following two: direction-of-arrival (DOA) estimation using array signal processing techniques and portfolio selection for financial asset management.

stat.ME↗

arXiv2022

Comments on "Iteratively Re-weighted Algorithm for Fuzzy c-Means"

In this comment, we present a simple alternate derivation to the IRW-FCM algorithm presented in "Iteratively Re-weighted Algorithm for Fuzzy c-Means" for Fuzzy c-Means problem. We show that the iterative steps derived for IRW-FCM algorithm are nothing but steps of the popular Majorization Minimization (MM) algorithm. The derivation presented in this note is much simpler and straightforward and, unlike the derivation of IRW-FCM, the derivation here does not involve introduction of any auxiliary variable. Moreover, by showing the steps of IRW-FCM as the MM algorithm, the inner loop of the IRW-FCM algorithm can be eliminated and the algorithm can be effectively run as a "single loop" algorithm. More precisely, the new MM-based derivation deduces that a single inner loop of IRW-FCM is sufficient to decrease the Fuzzy c-means objective function, which speeds up the IRW-FCM algorithm.

cs.AI↗

arXiv2022

Optimal Sensor Placement for Hybrid Source Localization Using Fused TOA-RSS-AOA Measurements

Source localization techniques incorporating hybrid measurements improve the reliability and accuracy of the location estimate. Given a set of hybrid sensors that can collect combined time of arrival (TOA), received signal strength (RSS) and angle of arrival (AOA) measurements, the localization accuracy can be enhanced further by optimally designing the placements of the hybrid sensors. In this paper, we present an optimal sensor placement methodology, which is based on the principle of majorization-minimization (MM), for hybrid localization technique. We first derive the Cramer-Rao lower bound (CRLB) of the hybrid measurement model, and formulate the design problem using the A-optimal criterion. Next, we introduce an auxiliary variable to reformulate the design problem into an equivalent saddle-point problem, and then construct simple surrogate functions (having closed form solutions) over both primal and dual variables. The application of MM in this paper is distinct from the conventional MM (that is usually developed only over the primal variable), and we believe that the MM framework developed in this paper can be employed to solve many optimization problems. The main advantage of our method over most of the existing state-of-the-art algorithms (which are mostly analytical in nature) is its ability to work for both uncorrelated and correlated noise in the measurements. We also discuss the extension of the proposed algorithm for the optimal placement designs based on D and E optimal criteria. Finally, the performance of the proposed method is studied under different noise conditions and different design parameters.

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arXiv2022

Designing Interference-Immune Doppler-TolerantWaveforms for Automotive Radar Applications

Dynamic target detection using FMCW waveform is challenging in the presence of interference for different radar applications. Degradation in SNR is irreparable and interference is difficult to mitigate in time and frequency domain. In this paper, a waveform design problem is addressed using the Majorization-Minimization (MM) framework by considering PSL/ISL cost functions, resulting in a code sequence with Doppler-tolerance characteristics of an FMCW waveform and interference immune characteristics of a tailored PMCW waveform (unique phase code + minimal ISL/PSL). The optimal design sequences possess polynomial phase behavior of degree Q amongst its sub-sequences and obtain optimal ISL and PSL solutions with guaranteed convergence. By tuning the optimization parameters such as degree Q of the polynomial phase behavior, sub-sequence length M and the total number of sub-sequences L, the optimized sequences can be as Doppler tolerant as FMCW waveform in one end, and they can possess small cross-correlation values similar to random-phase sequences in PMCW waveform on the other end. If required in the event of acute interference, new codes can be generated in the runtime which have low cross-correlation with the interferers. The performance analysis indicates that the proposed method outperforms the state-of-the-art counterparts.

cs.IT↗

arXiv2022

Optimal Sensor Placement for Source Localization: A Unified ADMM Approach

Source localization plays a key role in many applications including radar, wireless and underwater communications. Among various localization methods, the most popular ones are Time-Of-Arrival (TOA), Time-Difference-Of-Arrival (TDOA), and Received Signal Strength (RSS) based. Since the Cramér-Rao lower bounds (CRLB) of these methods depend on the sensor geometry explicitly, sensor placement becomes a crucial issue in source localization applications. In this paper, we consider finding the optimal sensor placements for the TOA, TDOA and RSS based localization scenarios. We first unify the three localization models by a generalized problem formulation based on the CRLB-related metric. Then a unified optimization framework for optimal sensor placement (UTMOST) is developed through the combination of the alternating direction method of multipliers (ADMM) and majorization-minimization (MM) techniques. Unlike the majority of the state-of-the-art works, the proposed UTMOST neither approximates the design criterion nor considers only uncorrelated noise in the measurements. It can readily adapt to to different design criteria (i.e. A, D and E-optimality) with slight modifications within the framework and yield the optimal sensor placements correspondingly. Extensive numerical experiments are performed to exhibit the efficacy and flexibility of the proposed framework.

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arXiv2022

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.

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arXiv2021

New Methods for MLE of Toeplitz Structured Covariance Matrices with Applications to RADAR Problems

This work considers Maximum Likelihood Estimation (MLE) of a Toeplitz structured covariance matrix. In this regard, an equivalent reformulation of the MLE problem is introduced and two iterative algorithms are proposed for the optimization of the equivalent problem. Both the strategies are based on the Majorization Minimization (MM) framework and hence enjoy nice properties such as monotonicity and ensured convergence to a stationary point of the equivalent MLE problem. The proposed algorithms are also extended to deal with MLE of other related covariance structures, namely, the banded Toeplitz, Toeplitz-block-Toeplitz, low rank Toeplitz structure plus a scalar matrix (accounting for white noise), and finally Toeplitz matrices satisfying a condition number constraint. Through numerical simulations, it is shown that new methods provide satisfactory performance levels in terms of both mean square estimation error and signal-to-interference-plus-noise ratio.

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arXiv2021

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.

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arXiv2021

UNIPOL: Unimodular sequence design via a separable iterative quartic polynomial optimization for active sensing systems

Sequences having better autocorrelation properties play a crucial role in enhancing the performance of active sensing systems. Hence, sequences with good autocorrelation properties are very much in demand. In this paper, we addressed the problem of designing a unimodular sequence having better side-lobe levels. We formulated it as a constrained optimization problem comprising the Integrated Side-lobe Level (ISL) metric and then proposed an effective algorithm (named UNIPOL - UNImodular sequence design via a separable iterative POLynomial optimization) where we perform the polynomial optimization at every iteration. The novelty of the paper comes from deriving a quartic majorization function that is separable in the sequence variables and that can be minimized parallelly. To evaluate the performance of our proposed algorithm we perform the numerical experiments for different sequence lengths and confirm that our proposed algorithm is the fastest algorithm to attain an actual optimum minimizer of the ISL metric. Our proposed algorithm is also computationally efficient due to its ease of implementation using the FFT, IFFT operations in a parallel fashion.

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arXiv2021

Design of MIMO Radar Waveforms based on lp-Norm Criteria

Multiple-input multiple-output (MIMO) radars transmit a set of sequences that exhibit small cross-correlation sidelobes, to enhance sensing performance by separating them at the matched filter outputs. The waveforms also require small auto-correlation sidelobes to avoid masking of weak targets by the range sidelobes of strong targets and to mitigate deleterious effects of distributed clutter. In light of these requirements, in this paper, we design a set of phase-only (constant modulus) sequences that exhibit near-optimal properties in terms of Peak Sidelobe Level (PSL) and Integrated Sidelobe Level (ISL). At the design stage, we adopt weighted lp-norm of auto- and cross-correlation sidelobes as the objective function and minimize it for a general p value, using block successive upper bound minimization (BSUM). Considering the limitation of radar amplifiers, we design unimodular sequences which make the design problem non-convex and NP-hard. To tackle the problem, in every iteration of the BSUM algorithm, we introduce different local approximation functions and optimize them concerning a block, containing a code entry or a code vector. The numerical results show that the performance of the optimized set of sequences outperforms the state-of-the-art counterparts, in both terms of PSL values and computational time.

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arXiv2020

Designing sequence set with minimal peak side-lobe level for applications in high resolution RADAR imaging

Constant modulus sequence set with low peak side-lobe level is a necessity for enhancing the performance of modern active sensing systems like Multiple Input Multiple Output (MIMO) RADARs. In this paper, we consider the problem of designing a constant modulus sequence set by minimizing the peak side-lobe level, which can be cast as a non-convex minimax problem, and propose a Majorization-Minimization technique based iterative monotonic algorithm. The iterative steps of our algorithm are computationally not very demanding and they can be efficiently implemented via Fast Fourier Transform (FFT) operations. We also establish the convergence of our proposed algorithm and discuss the computational and space complexities of the algorithm. Finally, through numerical simulations, we illustrate the performance of our method with the state-of-the-art methods. To highlight the potential of our approach, we evaluate the performance of the sequence set designed via our approach in the context of probing sequence set design for MIMO RADAR angle-range imaging application and show results exhibiting good performance of our method when compared with other commonly used sequence set design approaches.

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