LU decomposition and Toeplitz decomposition of a neural network

The universal approximation theorem asserts that a single hidden layer neural network approximates continuous functions with any desired precision on compact sets. As an existential result, the universal approximation theorem supports the use of neural networks for various applications, including regression and classification tasks. The universal approximation theorem is not limited to real-valued neural networks but also holds for complex, quaternion, tessarines, and Clifford-valued neural networks. This paper extends the universal approximation theorem for a broad class of hypercomplex-valued neural networks. Precisely, we first introduce the concept of non-degenerate hypercomplex algebra. Complex numbers, quaternions, and tessarines are examples of non-degenerate hypercomplex algebras. Then, we state the universal approximation theorem for hypercomplex-valued neural networks defined on a non-degenerate algebra.KeywordsHypercomplex algebrasNeural networksUniversal approximation theorem

Multiple LU factorizations of a singular matrix

In this paper, we propose a general projection neural network for solving a wider class of optimization and related problems. In addition to its simple structure and low complexity, the proposed neural network include existing neural networks for optimization, such as the projection neural network, the primal-dual neural network, and the dual neural network, as special cases. Under various mild conditions, the proposed general projection neural network is shown to be globally convergent, globally asymptotically stable, and globally exponentially stable. Furthermore, several improved stability criteria on two special cases of the general projection neural network are obtained under weaker conditions. Simulation results demonstrate the effectiveness and characteristics of the proposed neural network.

On the relative expressiveness of Bayesian and neural networks

Efficient parallel factorization and solution of structured and unstructured linear systems

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.

Universal approximation capability is an important research topic in artificial neural networks. The purpose of this study is to investigate universal approximation capability of a single hidden layer feed forward multiple 2π-periodic approximate identity neural networks in two function spaces. We present the notion of multiple 2π-periodic approximate identity. With respect to this notion, we prove two theorems in the space of continuous multivariate 2π-periodic functions. The second theorem shows that the above networks have universal approximation capability. The proof of the theorem uses a technique based on the notion of epsilon-net. Moreover, we discuss the universal approximation capability of the networks in the space of Lebesgue integrable multivariate 2π-periodic functions. The results of this study will be able to extend the standard theory of the universal approximation capability of feedforward neural networks.

Considering the features of long term load forecasting are complicated, this paper proposes a generic neural network model that is able to adapt to and learn from amount of non-linear or imprecise rules, so it is a model with highly robustness. For avoiding the inflexibility of the generic neural network itself, many experiences and opinions of experts are introduced during the use, so that a comprehensive effect of different factors that influence the power load can be reflected. The generic algorithm is able to search precisely at global scope, and the neural network is able to fit well at local scope, both of which are chosen together by this paper, i.e. the generic neural network algorithm. In the simulation training of the model, data from 1990 to 2007 of 16 indexes are used, and experience and opinions of national regulation factors of primary, secondary and tertiary industries that are provided by experts are also followed, and the long term load forecasting is carried out by rolling annual extrapolation.

In this paper, by employing the Zhang neural network (ZNN) method, an effective continuous-time LU decomposition (CTLUD) model is firstly proposed, analyzed, and investigated for solving the time-varying LU decomposition problem. Then, for the convenience of digital hardware realization, this paper proposes three discrete-time models by using Euler, 4-instant Zhang et al. discretization (ZeaD), and 8-instant ZeaD formulas to discretize the proposed CTLUD model, respectively. Furthermore, the proposed models are used to perform the LU decomposition of three time-varying matrices with different dimensions. Results indicate that the proposed models are effective for solving the time-varying LU decomposition problem, and the 8-instant ZeaD LU decomposition model has the highest precision among the three discrete-time models.

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.

One of the main computer-learning tools is an (artificial) neural network (NN); based on the values y (p) of a certain physical quantity y at several points x (p) =(x 1 (p) ,...,x n (p) ), the NN finds a dependence y = f(x1,...,x n ) that explains all known observations and predicts the value of y for other x = (x1,...,xn). The ability to describe an arbitrary dependence follows from the universal approximation theorem, according to which an arbitrary continuous function of a bounded set can be, within a given accuracy, approximated by an appropriate NN.

Nested range conditions for LU factorizations of integer matrices