Multiple LU factorizations of a singular matrix

LU decomposition and Toeplitz decomposition of a neural network

Semi-symmetric three-way arrays are essential tools in blind source separation (BSS) particularly in independent component analysis (ICA). These arrays can be built by resorting to higher order statistics of the data. The canonical polyadic (CP) decomposition of such semi-symmetric three-way arrays allows us to identify the so-called mixing matrix, which contains the information about the intensities of some latent source signals present in the observation channels. In addition, in many applications, such as the magnetic resonance spectroscopy (MRS), the columns of the mixing matrix are viewed as relative concentrations of the spectra of the chemical components. Therefore, the two loading matrices of the three-way array, which are equal to the mixing matrix, are nonnegative. Most existing CP algorithms handle the symmetry and the nonnegativity separately. Up to now, very few of them consider both the semi-nonnegativity and the semi-symmetry structure of the three-way array. Nevertheless, like all the methods based on line search, trust region strategies, and alternating optimization, they appear to be dependent on initialization, requiring in practice a multi-initialization procedure. In order to overcome this drawback, we propose two new methods, called JD LU + and JD QR + , to solve the problem of CP decomposition of semi-nonnegative semi-symmetric three-way arrays. Firstly, we rewrite the constrained optimization problem as an unconstrained one. In fact, the nonnegativity constraint of the two symmetric modes is ensured by means of a square change of variable. Secondly, a Jacobi-like optimization procedure is adopted because of its good convergence property. More precisely, the two new methods use LU and QR matrix factorizations, respectively, which consist in formulating high-dimensional optimization problems into several sequential polynomial and rational subproblems. By using both LU and QR matrix factorizations, we aim at studying the influence of the used matrix factorization. Numerical experiments on simulated arrays emphasize the advantages of the proposed methods especially the one based on LU factorization, in the presence of high-variance model error and of degeneracies such as bottlenecks. A BSS application on MRS data confirms the validity and improvement of the proposed methods.

When one column of a square nonsingular matrix, say $B^0 $, is replaced by another, the Bartels–Golub update of the LU factors of $B^0 $ forms a product form representation that is suitable for the efficient solution of associated systems of linear equations. However, it does not update the factors so as to obtain (even implicitly) the LU factors of the new matrix. In this note an exceedingly simple modification of the Bartels–Golub technique that yields the true LU factors is described, and it is shown to be mathematically equivalent to the update (specifically the partial pivoting version) recently proposed by Fletcher and Matthews [Math. Programming, 30 (1984), pp. 267–284]. The other commonly used update, Forrest–Tomlin, is quite evidently a mathematical variant of the Bartels–Golub technique, so this result enables a common ground for the three major updating techniques of linear programming to be established.

In Gaussian elimination and LU factorization we solve a linear system by transforming it to triangular form. These are not the only kind of transformations that can be used for such a task. Matrices with orthonormal columns, called unitary matrices can be used to reduce a square matrix to upper triangular form and more generally a rectangular matrix to upper triangular (also called upper trapezoidal) form. This lead to a decomposition of a rectangular matrix known as a QR decomposition and a reduced form which we refer to as a QR factorization. The QR decomposition and factorization will be used in later chapters to solve least squares- and eigenvalue problems.

In recent years, protecting copyright of digital images is an indispensable requirement for owners. To against with rapidly increasing of attacks, many techniques have been proposed in transform domain for ensuring quality of watermarked image, robustness of extracted watermark and execution time. Among these techniques, LU decomposition is considered as an outstanding technique in term of computation. However, it is that not all square matrices have an LU decomposition. Therefore, the suitable blocks need to be chosen before factorizing pixel matrices into lower and upper triangular matrix. In addition, in order to improve the invisibility of watermarked image, watermark should be embedded on one element of L matrix instead of two elements as the previous proposals. In this paper, we propose a novel image watermarking scheme which is based on strategy of LU blocks selection and an improved embedding method. Beside that, the extraction time is significantly sped up by a new solution to get out L(2,1) and L(3,1) elements of L matrix without performing LU decomposition in the extracting stage. According to the experimental results, our proposed method not only has the much better visual quality of watermarked images, but also can effectively extracts the watermark under some attacks.

Neutrosophic numbers and matrices extend classical fuzzy set theory by incorporating three components to represent uncertainty: truth membership, indeterminacy membership, and falsity membership degrees. Real neutrosophic matrices consist of neutrosophic numbers as their entries, offering a structure to model uncertain and imprecise information across various applications. This paper investigates operations on real neutrosophic matrices, such as addition, multiplication, and scalar multiplication, and defines how these operations function within the neutrosophic framework. The paper also examines the use of LU decomposition as an efficient method for solving systems of real neutrosophic linear algebraic equations. LU decomposition is a numerical technique that breaks down a matrix into lower and upper triangular matrices, enabling the solution of linear systems through forward and backward substitution.

Semi-fragile watermarking has been widely used for content authentication. In this paper, we propose a semi-fragile watermarking method using slant transform (ST) and LU decomposition (LUD) for image authentication. Initially, the watermark image is scrambled using an algorithm to enhance the confidentiality of the image. The host image is separated into red, green, and blue channels. The red channel is divided into 8×8 non-overlapping blocks. ST is applied to each block to get the ST coefficients. LU decomposition is then applied to these coefficients to obtain the lower and upper triangular matrix. Watermark data is embedded into the upper triangular matrix of ST coefficients using an embedding equation. Simulation results indicate that proposed method produces high quality watermarked image, provides secure embedding, extraction and authentication. Moreover, it effectively detects malicious tempering on image while being resistant to various content preserving manipulations. Furthermore, it shows superior performance than the state-of-the-art methods in terms of imperceptibility.

Matrix factorizations or matrix decompositions are methods that represent a matrix as a product of two or more matrices. There are various types of matrix factorizations such as LU factorization, Cholesky factorization, singular value decomposition etc. Matrix factorization is widely used in pattern recognition, image denoising, data clustering etc. Motivated by these applications, some properties and applications of various types of matrix factorizations are studied. One of the purposes of matrix factorization is to ease the computation. Thus, comparisons in term of computation time of various matrix factorizations in different areas are carried out.

Lighting calculations involving iteration or repetition present a problem of interest for many lighting tasks. Repeated calculation is needed, for example, in a design process for the best design evaluation or when a lighting system performance simulation is required A fast solution of the repeated calculation problem can be of special value for the computer-aided design of a lighting system. This paper discusses how the problem of repeated lighting calculations could be handled by the LU decomposition method. This method is commonly used by numerical mathematics for the solution of a single system of linear equations Ax = b, when there is more than one light-hand side b. To speed up the process of repeated solutions, the method involves decomposing a matrix A into an equivalent pair L and U, where L and U are lower and upper triangular matrices, respectively. The proposed LU decomposition method is compared with the traditional solution methods and its performance is shown. Examples of its use are presented to illustrate a possible field for the application of the method.

Nested range conditions for LU factorizations of integer matrices

We present algorithms to determine the number of nonzeros in each row and column of the factors of a sparse matrix, for both the QR factorization and the LU factorization with partial pivoting. The algorithms use only the nonzero structure of the input matrix, and run in time nearly linear in the number of nonzeros in that matrix. They may be used to set up data structures or schedule parallel operations in advance of the numerical factorization. The row and column counts we compute are upper bounds on the actual counts. If the input matrix is strong Hall and there is no coincidental numerical cancellation, the counts are exact for QR factorization and are the tightest bounds possible for LU factorization. These algorithms are based on our earlier work on computing row and column counts for sparse Cholesky factorization, plus an efficient method to compute the column elimination tree of a sparse matrix without explicitly forming the product of the matrix and its transpose.

Explicit factorization of the Vandermonde matrix

A recent approach for solving sparse triangular systems of equations on massively parallel computers employs a factorization of the triangular coefficient matrix to obtain a representation of its inverse in product form. The number of general communication steps required by this approach is proportional to the number of factors in the factorization. The triangular matrix can be symmetrically permuted to minimize the number of factors over suitable classes of permutations, and thereby the complexity of the parallel algorithm can be minimized. Algorithms for minimizing the number of factors over several classes of permutations have been considered in earlier work. Let F = L+L[sup T] denote the symmetric filled matrix corresponding to a Cholesky factor L, and let G[sub F] denote the adjacency graph of F. In this paper we consider the problem of minirriizing the number of factors over all permutations which preserve the structure of G[sub F]. The graph model of this problem is to partition the vertices G[sub F] into the fewest transitively closed subgraphs over all perfect elimination orderings while satisfying a certain precedence relationship. The solution to this chordal graph partitioning problem can be described by a greedy scheme which eliminates a largest permissible subgraph at each step. Further, the subgraph eliminated at each step can be characterized in terms of lengths of chordless paths in the current elimination graph. This solution relies on several results concerning transitive perfect elimination orderings introduced in this paper. We describe a partitioning algorithm with [Omicron]([vert bar]V[vert bar] + [vert bar]E[vert bar]) time and space complexity.

The backstepping control design algorithms described by Krstic et al. (1995) provide a systematic framework for the design of regulating strategies suitable for large classes of nonlinear uncertain systems. However, the equations arising at the successive steps are usually too complicated to be computed by hand. We consider here a symbolic toolbox which implements a general algorithm for the design of dynamic adaptive controllers following the basic ideas of backstepping with tuning functions without transformation into canonical forms. This algorithm is applicable to observable minimum phase systems not necessarily in triangular form and also to uncertain nonlinear systems in triangular forms. Additionally the control can be generated by a sliding mode approach.

Recent results have shown that topology aware mapping reduces network contention in communication-intensive kernels on massively parallel machines. We demonstrate that on mesh interconnects, topology aware mapping also allows for the utilization of highly-efficient topology aware collectives. We map novel 2.5D dense linear algebra algorithms to exploit rectangular collectives on cuboid partitions allocated by a Blue Gene/P supercomputer. Our mappings allow the algorithms to exploit optimized line multicasts and reductions. Commonly used 2D algorithms cannot be mapped in this fashion. On 16,384 nodes (65,536 cores) of Blue Gene/P, 2.5D algorithms that exploit rectangular collectives are significantly faster than 2D matrix multiplication (MM) and LU factorization, up to 8.7x and 2.1x, respectively. These speed-ups are due to communication reduction (up to 95.6% for 2.5D MM with respect to 2D MM). We also derive LogP-based novel performance models for rectangular broadcasts and reductions. Using those, we model the performance of matrix multiplication and LU factorization on a hypothetical exascale architecture.