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Erna Begovic

Publications and source records attributed to Erna Begovic.

16 recordsLinked to original sources
arXiv2026

Convergence and mixed-precision preconditioning for the naive Jacobi eigenvalue algorithm

The paper studies a Jacobi-type method for the eigenvalue problem of general complex matrices with simple eigenvalues. The method applies elementary triangular similarity transformations in order to annihilate selected off-diagonal elements and, when convergent, produces highly accurate eigenvalues. We give a new proof of its asymptotic quadratic convergence and derive an explicit, verifiable bound that describes the region in which this convergence is guaranteed. To make the method applicable well beyond matrices already close to the diagonal form, we introduce a preconditioning strategy. We use two types of preconditioners, both based on theoretical convergence results. The preconditioner is computed at lower precision to reduce computational cost, the associated similarity transformation is applied either at working or at higher precision, to preserve spectral information, while the main algorithm performs at working precision. Numerical experiments demonstrate that the resulting algorithm is robust and produces very accurate eigenvalues.

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arXiv2025

On the block Eberlein diagonalization method

The Eberlein diagonalization method is an iterative Jacobi-type method for solving the eigenvalue problem of a general complex matrix. In this paper we develop the block version of the Eberlein method. We prove the global convergence of our block method and present several numerical examples.

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arXiv2024

Randomized coupled decompositions

Coupled decompositions are a widely used tool for data fusion. As the volume of data increases, so does the dimensionality of matrices and tensors, highlighting the need for more efficient coupled decomposition algorithms. This paper studies the problem of coupled matrix factorization (CMF), where two matrices represented in low-rank form share a common factor. Additionally, it explores coupled matrix and tensor factorization (CMTF), where a matrix and a tensor are represented in low-rank form, also sharing a common factor matrix. We show that these problems can be solved using a direct approach with singular value decomposition (SVD), rather than relying on an iterative method. Knowing that matrices coming from real-world applications are often very large, the computational cost can be substantial. To address this issue and improve the efficiency, we propose new techniques for randomizing these algorithms. This includes a novel strategy for selecting a projection subspace that takes into account the contribution from both matrices involved in the decomposition equally. We present extensive results of numerical tests that confirm the efficiency of our algorithms. Furthermore, as a novel approach and with a high success rate, we apply our randomized algorithms to the face recognition problem.

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arXiv2023

Convergence of the complex block Jacobi methods under the generalized serial pivot strategies

The paper considers the convergence of the complex block Jacobi diagonalization methods under the large set of the generalized serial pivot strategies. The global convergence of the block methods for Hermitian, normal and $J$-Hermitian matrices is proven. In order to obtain the convergence results for the block methods that solve other eigenvalue problems, such as the generalized eigenvalue problem, we consider the convergence of a general block iterative process which uses the complex block Jacobi annihilators and operators.

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arXiv2022

CP decomposition and low-rank approximation of antisymmetric tensors

For the antisymmetric tensors the paper examines a low-rank approximation which is represented via only three vectors. We describe a suitable low-rank format and propose an alternating least squares structure-preserving algorithm for finding such approximation. Moreover, we show that this approximation problem is equivalent to the problem of finding the best multilinear low-rank antisymmetric approximation and, consequently, equivalent to the problem of finding the best unstructured rank-$1$ approximation. The case of partial antisymmetry is also discussed. The algorithms are implemented in Julia programming language and their numerical performance is discussed.

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arXiv2022

Convergence of the Eberlein diagonalization method under the generalized serial pivot strategies

The Eberlein method is a Jacobi-type process for solving the eigenvalue problem of an arbitrary matrix. In each iteration two transformations are applied on the underlying matrix, a plane rotation and a non-unitary elementary transformation. The paper studies the method under the broad class of generalized serial pivot strategies. We prove the global convergence of the Eberlein method under the generalized serial pivot strategies with permutations and present several numerical examples.

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arXiv2021

Trace maximization algorithm for the approximate tensor diagonalization

In this paper we develop a Jacobi-type algorithm for the approximate diagonalization of tensors of order $d\geq3$ via tensor trace maximization. For a general tensor this is an alternating least squares algorithm and the rotation matrices are chosen in each mode one-by-one to maximize the tensor trace. On the other hand, for symmetric tensors we discuss a structure-preserving variant of this algorithm where in each iteration the same rotation is applied in all modes. We show that both versions of the algorithm converge to the stationary points of the corresponding objective functions.

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arXiv2021

Convergence of a Jacobi-type method for the approximate orthogonal tensor diagonalization

For a general third-order tensor $\mathcal{A}\in\mathbb{R}^{n\times n\times n}$ the paper studies two closely related problems, an SVD-like tensor decomposition and an (approximate) tensor diagonalization. We develop a Jacobi-type algorithm that works on $2\times2\times2$ subtensors and, in each iteration, maximizes the sum of squares of its diagonal entries. We show how the rotation angles are calculated and prove convergence of the algorithm. Different initializations of the algorithm are discussed, as well as the special cases of symmetric and antisymmetric tensors. The algorithm can be generalized to work on higher-order tensors.

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arXiv2020

Finding the closest normal structured matrix

Given a structured matrix $A$ we study the problem of finding the closest normal matrix with the same structure. The structures of our interest are: Hamiltonian, skew-Hamiltonian, per-Hermitian, and perskew-Hermitian. We develop a structure-preserving Jacobi-type algorithm for finding the closest normal structured matrix and show that such algorithm converges to a stationary point of the objective function.

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arXiv2020

Hybrid CUR-type decomposition of tensors in the Tucker format

The paper introduces a hybrid approach to the CUR-type decomposition of tensors in the Tucker format. The idea of the hybrid algorithm is to write a tensor $\mathcal{X}$ as a product of a core tensor $\mathcal{S}$, a matrix $C$ obtained by extracting mode-$k$ fibers of $\mathcal{X}$, and matrices $U_j$, $j=1,\ldots,k-1,k+1,\ldots,d$, chosen to minimize the approximation error. The approximation can easily be modified to preserve the fibers in more than one mode. The approximation error obtained this way is smaller than the one from the standard tensor CUR-type method. This difference increases as the tensor dimension increases. It also increases as the number of modes in which the original fibers are preserved decreases.

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arXiv2018

On the convergence of complex Jacobi methods

In this paper we prove the global convergence of the complex Jacobi method for Hermitian matrices for a large class of generalized serial pivot strategies. For a given Hermitian matrix $A$ of order $n$ we find a constant $\gamma<1$ depending on $n$, such that $S(A')\leq\gamma{S(A)}$, where $A'$ is obtained from $A$ by applying one or more cycles of the Jacobi method and $S(\cdot)$ stands for the off-norm. Using the theory of complex Jacobi operators, the result is generalized so it can be used for proving convergence of more general Jacobi-type processes. In particular, we use it to prove the global convergence of Cholesky-Jacobi method for solving the positive definite generalized eigenvalue problem.

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arXiv2017

Jacobi method for symmetric $4\times4$ matrices converges for every cyclic pivot strategy

The paper studies the global convergence of the Jacobi method for symmetric matrices of size $4$. We prove global convergence for all $720$ cyclic pivot strategies. Precisely, we show that inequality $S(A^{[t+3]})\leq\gamma S(A^{[t]})$, $t\geq1$, holds with the constant $\gamma<1$ that depends neither on the matrix $A$ nor on the pivot strategy. Here $A^{[t]}$ stands for the matrix obtained from $A$ after $t$ full cycles of the Jacobi method and $S(A)$ is the off-diagonal norm of $A$. We show why three consecutive cycles have to be considered. The result has a direct application on the $J$-Jacobi method.

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arXiv2017

On the global convergence of the Jacobi method for symmetric matrices of order 4 under parallel strategies

The paper analyzes special cyclic Jacobi methods for symmetric matrices of order $4$. Only those cyclic pivot strategies that enable full parallelization of the method are considered. These strategies, unlike the serial pivot strategies, can force the method to be very slow or very fast within one cycle, depending on the underlying matrix. Hence, for the global convergence proof one has to consider two or three adjacent cycles. It is proved that for any symmetric matrix $A$ of order~$4$ the inequality $S(A^{[2]})\leq(1-10^{-5})S(A)$ holds, where $A^{[2]}$ results from $A$ by applying two cycles of a particular parallel method. Here $S(A)$ stands for the Frobenius norm of the strictly upper-triangular part of $A$. The result holds for two special parallel strategies and implies the global convergence of the method under all possible fully parallel strategies. It is also proved that for every $\epsilon>0$ and $n\geq4$ there exist a symmetric matrix $A(\epsilon)$ of order $n$ and a cyclic strategy, such that upon completion of the first cycle of the appropriate Jacobi method the inequality $S(A^{[1]})> (1-\epsilon)S(A(\epsilon))$ holds.

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arXiv2016

Convergence of the Cyclic and Quasi-cyclic Block Jacobi Methods

The paper studies the global convergence of the block Jacobi me\-thod for symmetric matrices. Given a symmetric matrix $A$ of order $n$, the method generates a sequence of matrices by the rule $A^{(k+1)}=U_k^TA^{(k)}U_k$, $k\geq0$, where $U_k$ are orthogonal elementary block matrices. A class of generalized serial pivot strategies is introduced, significantly enlarging the known class of weak wavefront strategies, and appropriate global convergence proofs are obtained. The results are phrased in the stronger form: $S(A')\leq c S(A)$, where $A'$ is the matrix obtained from $A$ after one full cycle, $c<1$ is a constant and $S(A)$ is the off-norm of $A$. Hence, using the theory of block Jacobi operators, one can apply the obtained results to prove convergence of block Jacobi methods for other eigenvalue problems, such as the generalized eigenvalue problem. As an example, the results are applied to the block $J$-Jacobi method. Finally, all results are extended to the corresponding quasi-cyclic strategies.

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arXiv2016

Structure-preserving low multilinear rank approximation of antisymmetric tensors

This paper is concerned with low multilinear rank approximations to antisymmetric tensors, that is, multivariate arrays for which the entries change sign when permuting pairs of indices. We show which ranks can be attained by an antisymmetric tensor and discuss the adaption of existing approximation algorithms to preserve antisymmetry, most notably a Jacobi algorithm. Particular attention is paid to the important special case when choosing the rank equal to the order of the tensor. It is shown that this case can be addressed with an unstructured rank-$1$ approximation. This allows for the straightforward application of the higher-order power method, for which we discuss effective initialization strategies.

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