arXiv subjects

Nezam Mahdavi-Amiri

Publications and source records attributed to Nezam Mahdavi-Amiri.

8 recordsLinked to original sources

arXiv2026

Exploiting the the nonzero diagonal pattern in matrix function computations

We consider the task of approximating a matrix function $f(A)$, where $A$ is a matrix in which only a relatively small number of (not necessarily consecutive) sub- and superdiagonals contain nonzero entries. Approximating $f$ by a low-degree polynomial $p$ allows us to obtain sparse approximations to $f(A)$, which one can efficiently work with (while, in general, $f(A)$ is a dense matrix, even when $A$ is sparse). Our approach is based on carefully inspecting the locations where nonzeros can occur in $p(A)$, and identifying the entries in $A$ that influence them. In particular, we illustrate how this approach can be used for efficiently approximating the trace of $f(A)$ and identify how this approach is related to established (stochastic) probing methods for trace estimation. Another application area in which our approach works particularly well is the computation of functions of Toeplitz matrices. Here, studying the sparsity pattern of $p(A)$ allows us to reduce the computation of the whole matrix polynomial to that of a single small-scale submatrix, yielding an algorithm that scales exceptionally well to large problem sizes.

math.NA↗

arXiv2021

Image Enhancement via Bilateral Learning

Nowadays, due to advanced digital imaging technologies and internet accessibility to the public, the number of generated digital images has increased dramatically. Thus, the need for automatic image enhancement techniques is quite apparent. In recent years, deep learning has been used effectively. Here, after introducing some recently developed works on image enhancement, an image enhancement system based on convolutional neural networks is presented. Our goal is to make an effective use of two available approaches, convolutional neural network and bilateral grid. In our approach, we increase the training data and the model dimensions and propose a variable rate during the training process. The enhancement results produced by our proposed method, while incorporating 5 different experts, show both quantitative and qualitative improvements as compared to other available methods.

eess.IV↗

arXiv2019

Nonconvex weak sharp minima on Riemannian manifolds

We are to establish necessary conditions (of the primal and dual types) for the set of weak sharp minima of a nonconvex optimization problem on a Riemannian manifold. Here, we are to provide a generalization of some characterizations of weak sharp minima for convex problems on Riemannian manifold introduced by Li et al. (SIAM J. Optim., 21 (2011), pp. 1523--1560) for nonconvex problems. We use the theory of the Fréchet and limiting subdifferentials on Riemannian manifold to give the necessary conditions of the dual type. We also consider a theory of contingent directional derivative and a notion of contingent cone on Riemannian manifold to give the necessary conditions of the primal type. Several definitions have been provided for the contingent cone on Riemannian manifold. We show that these definitions, with some modifications, are equivalent. We establish a lemma about the local behavior of a distance function. Using the lemma, we express the Fréchet subdifferential (contingent directional derivative) of a distance function on a Riemannian manifold in terms of normal cones (contingent cones), to establish the necessary conditions. As an application, we show how one can use weak sharp minima property to model a Cheeger type constant of a graph as an optimization problem on a Stiefel manifold.

math.OC↗

arXiv2014

Computing Symmetric Positive Definite Solutions of Three Types of Nonlinear Matrix Equations

Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute a symmetric and positive definite solution. Here, we propose new iterative algorithms for solving three different types of nonlinear matrix equations. We have recently proposed a new algorithm for solving positive definite total least squares problems. Making use of an iterative process for inverse of a matrix, we convert the nonlinear matrix equation to an iterative linear one, and, in every iteration, we apply our algorithm for solving a positive definite total least squares problem to solve the linear subproblem and update the newly defined variables and the matrix inverse terms using appropriate formulas. Our proposed algorithms have a number of useful features. One is that the computed unknown matrix remains symmetric and positive definite in all iterations. As the second useful feature, numerical test results show that in most cases our proposed approach turns to compute solutions with smaller errors within lower computing times. Finally, we provide some test results showing that our proposed algorithm converges to a symmetric and positive definite solution in Matlab software environment on a PC, while other methods fail to do so.

math.NA↗

arXiv2014

Efficient Algorithms for Positive Semi-Definite Total Least Squares Problems, Minimum Rank Problem and Correlation Matrix Computation

We have recently presented a method to solve an overdetermined linear system of equations with multiple right hand side vectors, where the unknown matrix is to be symmetric and positive definite. The coefficient and the right hand side matrices are respectively named data and target matrices. A more complicated problem is encountered when the unknown matrix is to be positive semi-definite. The problem arises in estimating the compliance matrix to model deformable structures and approximating correlation and covariance matrices in financial modeling. Several methods have been proposed for solving such problems assuming that the data matrix is unrealistically error free. Here, considering error in measured data and target matrices, we propose a new approach to solve a positive semi-definite constrained total least squares problem. We first consider solving the problem when the rank of the unknown matrix is known, by defining a new error formulation for the positive semi-definite total least squares problem and use of optimization methods on Stiefel manifolds. We prove quadratic convergence of our proposed approach. We then describe how to generalize our proposed method to solve the general positive semi-definite total least squares problem. We further apply the proposed approach to solve the minimum rank problem and the problem of computing correlation matrix. Comparative numerical results show the efficiency of our proposed algorithms. Finally, the Dolan-More performance profiles are shown to summarize our comparative study.

math.OC↗

arXiv2014

A note on "A simple algorithm to search for all MCs in networks"

Recently, Yeh [Yeh, WC. (2006). A simple algorithm to search for all MCs in networks. European Journal of Operational Research, 174, 1694{1705.] has proposed a simple algorithm to find all the Minimal Cuts in an undirected graph. However, the algorithm does not work properly. Here, using an example, a defect of this algorithm is illustrated, and then the corresponding result is shown to be incorrect. Moreover, a correct version of the algorithm is established.

math.OC↗

arXiv2014

Technical Notes on "A new approach to the d-MC problem"

System reliability is the probability of the maximum flow in a stochastic-flow network from the source node to the sink node being more than a demand level d. There are several approaches to compute system reliability using upper boundary points, called d-MinCuts (d-MCs). Search for all the d-MCs in a stochastic-flow network is an NP-hard problem. Here, a work proposed by Yeh [Yeh WC. A new approach to the d-MC problem. Reliab Eng and Syst Saf 2002; 77(2): 201-206.] for determining all the d-MCs is investigated. Two results (Lemma 3 and Theorem 5) are shown to be incorrect and their correct versions are established. Also, the complexity result (Theorem 6) is shown to be incorrect and the correct count is provided.

math.OC↗

arXiv2002

ABS Methods and ABSPACK for Linear Systems and Optimization, a Review

ABS methods are a large class of methods, based upon the Egervary rank reducing algebraic process, first introduced in 1984 by Abaffy, Broyden and Spedicato for solving linear algebraic systems, and later extended to nonlinear algebraic equations, to optimization problems and other fields; software based upon ABS methods is now under development. Current ABS literature consists of about 400 papers. ABS methods provide a unification of several classes of classical algorithms and more efficient new solvers for a number of problems. In this paper we review ABS methods for linear systems and optimization, from both the point of view of theory and the numerical performance of ABSPACK.

astro-ph↗