Prabhu Babu

Astha Saini, Prabhu Babu

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.

Ghania Fatima, Aakash Arora, Prabhu Babu et al.

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.

Konstantinos Benidis, Ying Sun, Prabhu Babu et al.

The problem of estimating sparse eigenvectors of a symmetric matrix attracts a lot of attention in many applications, especially those with high dimensional data set. While classical eigenvectors can be obtained as the solution of a maximization problem, existing approaches formulate this problem by adding a penalty term into the objective function that encourages a sparse solution. However, the resulting methods achieve sparsity at the expense of sacrificing the orthogonality property. In this paper, we develop a new method to estimate dominant sparse eigenvectors without trading off their orthogonality. The problem is highly non-convex and hard to handle. We apply the MM framework where we iteratively maximize a tight lower bound (surrogate function) of the objective function over the Stiefel manifold. The inner maximization problem turns out to be a rectangular Procrustes problem, which has a closed form solution. In addition, we propose a method to improve the covariance estimation problem when its underlying eigenvectors are known to be sparse. We use the eigenvalue decomposition of the covariance matrix to formulate an optimization problem where we impose sparsity on the corresponding eigenvectors. Numerical experiments show that the proposed eigenvector extraction algorithm matches or outperforms existing algorithms in terms of support recovery and explained variance, while the covariance estimation algorithms improve significantly the sample covariance estimator.

Ying Sun, Prabhu Babu, Daniel P. Palomar

This paper considers the problem of robustly estimating a structured covariance matrix with an elliptical underlying distribution with known mean. In applications where the covariance matrix naturally possesses a certain structure, taking the prior structure information into account in the estimation procedure is beneficial to improve the estimation accuracy. We propose incorporating the prior structure information into Tyler's M-estimator and formulate the problem as minimizing the cost function of Tyler's estimator under the prior structural constraint. First, the estimation under a general convex structural constraint is introduced with an efficient algorithm for finding the estimator derived based on the majorization minimization (MM) algorithm framework. Then, the algorithm is tailored to several special structures that enjoy a wide range of applications in signal processing related fields, namely, sum of rank-one matrices, Toeplitz, and banded Toeplitz structure. In addition, two types of non-convex structures, i.e., the Kronecker structure and the spiked covariance structure, are also discussed, where it is shown that simple algorithms can be derived under the guidelines of MM. Numerical results show that the proposed estimator achieves a smaller estimation error than the benchmark estimators at a lower computational cost.

Junxiao Song, Prabhu Babu, Daniel P. Palomar

In this paper, we consider an $\ell_{0}$-norm penalized formulation of the generalized eigenvalue problem (GEP), aimed at extracting the leading sparse generalized eigenvector of a matrix pair. The formulation involves maximization of a discontinuous nonconcave objective function over a nonconvex constraint set, and is therefore computationally intractable. To tackle the problem, we first approximate the $\ell_{0}$-norm by a continuous surrogate function. Then an algorithm is developed via iteratively majorizing the surrogate function by a quadratic separable function, which at each iteration reduces to a regular generalized eigenvalue problem. A preconditioned steepest ascent algorithm for finding the leading generalized eigenvector is provided. A systematic way based on smoothing is proposed to deal with the "singularity issue" that arises when a quadratic function is used to majorize the nondifferentiable surrogate function. For sparse GEPs with special structure, algorithms that admit a closed-form solution at every iteration are derived. Numerical experiments show that the proposed algorithms match or outperform existing algorithms in terms of computational complexity and support recovery.