Multilinear Algebra

This document summarizes the contributions of the Electromagnetic $\gamma_vNN^*$ Transition Form Factors workshop participants that provide theoretical support of the excited baryon program at the 12 GeV energy upgrade at JLab. The main objectives of the workshop were (a) review the status of the $\gamma_vNN^*$ transition form factors extracted from the meson electroproduction data, (b) call for the theoretical interpretations of the extracted $N$-$N^*$ transition form factors, that enable access to the mechanisms responsible for the N* formation and to their emergence from QCD.

This work presents an investigation into the mathematical structure of magic squares via unconventional "structure of discovery" of the document. The central finding is that magic squares form a complete, commutative, and associative algebra. This novel algebraic system is defined through the spectral decomposition of a unique permutation operator.  This discovery provides a general method for the construction of all magic squares of order 3 and larger from an already existing example, defines their operations and the geometric space they occupy. This document follows a narrative of discovery, developing its conclusions from the ground up rather than through a broad survey of existing literature.

Author is an artist, sharing the work in hope it will inspire further, more rigorous research. Python scripts were generated with AI and the parsing of the final proofs was AI assisted.

In this paper the symmetric differential and symmetric Lie deriv- ative are introduced. Using these tools derivations of the algebra of symmetric tensors are classified. We also define a Frolicher-Nijenhuis bracket for vector valued symmetric tensors.

We use the imbedding of the total differential operator D into a Heisenberg algebra to give a method to generate the transvectants and their multilinear generalizations using the coherent state method. This leads to tensor product decompositions where all the components play an equal role.

The widespread use of multisensor technology and the emergence of big datasets have created the need to develop tools to reduce, approximate, and classify large and multimodal data such as higher-order tensors. While early approaches focused on matrix and vector based methods to represent these higher-order data, more recently it has been shown that tensor decomposition methods are better equipped to capture couplings across their different modes. For these reasons, tensor decomposition methods have found applications in many different signal processing problems including dimensionality reduction, signal separation, linear regression, feature extraction, and classification. However, most of the existing tensor decomposition methods are based on the principle of finding a low-rank approximation in a linear subspace structure, where the definition of the rank may change depending on the particular decomposition. Since many datasets are not necessarily low-rank in a linear subspace, this often results in high approximation errors or low compression rates. In this paper, we introduce a new adaptive, multi-scale tensor decomposition method for higher order data inspired by hybrid linear modeling and subspace clustering techniques. In particular, we develop a multi-scale higher-order singular value decomposition (MS-HoSVD) approach where a given tensor is first permuted and then partitioned into several sub-tensors each of which can be represented as a low-rank tensor with increased representational efficiency. The proposed approach is evaluated for dimensionality reduction and classification for several different real-life tensor signals with promising results.

In this paper we present an efficiently scaling quantum algorithm which finds the size of the maximum common edge subgraph for a pair of arbitrary graphs and thus provides a meaningful measure of graph similarity. The algorithm makes use of a two-part quantum dynamic process: in the first part we obtain information crucial for the comparison of two graphs through linear quantum computation. However, this information is hidden in the quantum system with vanishingly small amplitude that even quantum algorithms such as Grover's search are not fast enough to distill the information efficiently. In order to extract the information we call upon techniques in nonlinear quantum computing to provide the speed-up necessary for an efficient algorithm. The linear quantum circuit requires O(n 3 log 3 (n) log log(n)) elementary quantum gates and the nonlinear evolution under the Gross-Pitaevskii equation has a time scaling of O( g n 2 log 3 (n) log log(n)), where n is the number of vertices in each graph and g is the strength of the Gross-Pitaveskii non-linearity. Through this example, we demonstrate the power of nonlinear quantum search techniques to solve a subset of NP-hard problems.

This paper deals with the singular configurations of symmetric 5-DOF parallel mechanisms performing three translational and two independent rotational DOFs. The screw theory approach is adopted in order to obtain the Jacobian matrices. The regularity of these matrices is examined using Grassmann-Cayley algebra and Grassmann geometry. More emphasis is placed on the geometric investigation of singular configurations by means of Grassmann-Cayley Algebra for a class of simplified designs whereas Grassmann geometry is used for a matter of comparison. The results provide algebraic expressions for the singularity conditions, in terms of some bracket monomials obtained from the superbracket decomposition. Accordingly, all the singularity conditions can be enumerated.

developed a more detailed analysis in the second part of my PhD thesis [Cantù 2003, 153-345], where the accent was put on the discontinuities between the two editions, and on Grassmann's criticism of the 'traditional' definition of mathematics as a theory of magnitudes. 5 "I define as an extensive magnitude any expression that is derived from a system of units (none of which need to be the absolute unit) by numbers, and I call the numbers that belong to the units the derivation numbers of that magnitude; for example the polynomial a 1 e 1 + a 2 e 2 + … or Σ ae or Σ r r a e , where a 1 , a 2 , … are real

We explore the constraints imposed by Poincaré duality on the resonance varieties of a graded algebra. For a three-dimensional Poincaré duality algebra A, we obtain a fairly precise geometric description of the resonance varieties ${\cal R}^i_k(A)$.

Every finitely presented group π can be realized as π = π 1 (M), for some smooth, compact, connected manifold M n of dim n ¥ 4. M n can be chosen to be orientable. If n even, n ¥ 4, then M n can be chosen to be symplectic (Gompf). If n even, n ¥ 6, then M n can be chosen to be complex (Taubes). Requiring that n = 3 puts severe restrictions on the (closed) 3-manifold group π = π 1 (M 3 ). A Kähler manifold is a compact, connected, complex manifold, with a Hermitian metric h such that ω = im(h) is a closed 2-form. The class of Kähler manifolds is closed under finite direct products, and passing to finite covers. Examples: smooth, complex projective varieties, such as Riemann surfaces. If M is a Kähler manifold, π = π 1 (M) is called a Kähler group. This also puts strong restrictions on G, e.g.: b 1 (π) is even (Hodge theory) π is 1-formal: Malcev Lie algebra m(G) is quadratic (DGMS 1975) π cannot split non-trivially as a free product (Gromov 1989) π finite ñ π projective group (Serre 1958). ALEX SUCIU (NORTHEASTERN) COMPLEX GEOMETRY AND 3-D TOPOLOGY GTA MINI-WORKSHOP 4 / 25 OVERVIEW KÄHLER GROUPS KÄHLER 3-MANIFOLD GROUPS QUESTION (DONALDSON-GOLDMAN 1989, REZNIKOV 1993) Which 3-manifold groups are Kähler groups? Reznikov (2002) gave a partial solution. Complete answer: THEOREM (DIMCA-S. 2009) Let π be the fundamental group of a closed 3-manifold. Then π is a Kähler group ðñ G is a finite subgroup of O(4), acting freely on S 3 . Alternative proofs have since been given by Kotschick (2012) and by Biswas, Mj and Seshadri (2012).

Single-word vector space models have been very successful at learning lexical information. However, they cannot capture the compositional meaning of longer phrases, preventing them from a deeper understanding of language. We introduce a recursive neural network (RNN) model that learns compositional vector representations for phrases and sentences of arbitrary syntactic type and length. Our model assigns a vector and a matrix to every node in a parse tree: the vector captures the inherent meaning of the constituent, while the matrix captures how it changes the meaning of neighboring words or phrases. This matrix-vector RNN can learn the meaning of operators in propositional logic and natural language. The model obtains state of the art performance on three different experiments: predicting fine-grained sentiment distributions of adverb-adjective pairs; classifying sentiment labels of movie reviews and classifying semantic relationships such as cause-effect or topic-message between nouns using the syntactic path between them.

Special state variable and transfer function descriptions are developed for systems whose unknown parameters are a wide class of physical element values. The dependence of the state variable structures on these parameters is "rank-1" and that of the numerator and denominator polynomials of the tranSfer functions multilinear.

We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or the spectral norm; and determining the rank or best rank-1 approximation of a 3-tensor. Furthermore, we show that restricting these problems to symmetric tensors does not alleviate their NP-hardness. We also explain how deciding nonnegative definiteness of a symmetric 4-tensor is NP-hard and how computing the combinatorial hyperdeterminant of a 4-tensor is NP-, #P-, and VNP-hard. We shall argue that our results provide another view of the boundary separating the computational tractability of linear/convex problems from the intractability of nonlinear/nonconvex ones.

Tensor decomposition is a powerful framework for multiway data analysis. Despite the success of existing approaches, they ignore the sparse nature of the tensor data in many real-world applications, explicitly or implicitly assuming dense tensors. To address this model misspecification and to exploit the sparse tensor structures, we propose Nonparametric dEcomposition of Sparse Tensors (NEST), which can capture both the sparse structure properties and complex relationships between the tensor nodes to enhance the embedding estimation. Specifically, we first use completely random measures to construct tensor-valued random processes. We prove that the entry growth is much slower than that of the corresponding tensor size, which implies sparsity. Given finite observations (i.e., projections), we then propose two nonparametric decomposition models that couple Dirichlet processes and Gaussian processes to jointly sample the sparse entry indices and the entry values (the latter as a nonlinear mapping of the embeddings), so as to encode both the structure properties and nonlinear relationships of the tensor nodes into the embeddings. Finally, we use the stick-breaking construction and random Fourier features to develop a scalable, stochastic variational learning algorithm. We show the advantage of our approach in sparse tensor generation, and entry index and value prediction in several real-world applications.

Tensor decomposition is a powerful framework for multiway data analysis. Despite the success of existing approaches, they ignore the sparse nature of the tensor data in many real-world applications, explicitly or implicitly assuming dense tensors. To address this model misspecification and to exploit the sparse tensor structures, we propose Nonparametric dEcomposition of Sparse Tensors (NEST), which can capture both the sparse structure properties and complex relationships between the tensor nodes to enhance the embedding estimation. Specifically, we first use completely random measures to construct tensor-valued random processes. We prove that the entry growth is much slower than that of the corresponding tensor size, which implies sparsity. Given finite observations (i.e., projections), we then propose two nonparametric decomposition models that couple Dirichlet processes and Gaussian processes to jointly sample the sparse entry indices and the entry values (the latter as a nonlin...

We show that the classical algebra of quaternions is a commutative Z 2 ×Z 2 ×Z 2 -graded algebra. A similar interpretation of the algebra of octonions is impossible. This note is our "private investigation" of what really happened on the 16th of October, 1843 on the Brougham Bridge when Sir William Rowan Hamilton engraved on a stone his fundamental relations: Since then, the elements i, j and k, together with the unit, 1, denote the canonical basis of the celebrated 4-dimensional associative algebra of quaternions H. Of course, the algebra H is not commutative: the relations above imply that the elements i, j, k anticommute with each other, for instance Yes, but...

In this paper, we propose some new preconditioners for solving multilinear system Axm-1 = b. These preconditioners are based on tensor splitting. We also present some theorems for analyzing and converging the preconditioned Jacobi, Gauss-Seidel, and SOR-type iterative methods. Numerical examples are presented to verify the efficiency of the proposed preconditioned methods. Mathematics Subject Classification (2010). 15A10, 15A69, 15A72, 15A99, 65F10.

This paper investigates a type of fast and flexible preconditioners to solve multilinear system Ax = b with M-tensor A and obtains some important convergent theorems about preconditioned Jacobi, GaussSeidel and SOR type iterative methods. The main results theoretically prove that the preconditioners can accelerate the convergence of iterations. Numerical examples are presented to reverify the efficiency of the proposed preconditioned methods.

i ii Ao Barros e à Laura, que deram início a tudo isso. Ao Barros Neto e à Ana Carolina pelo apoio. Ao Bernardo, meu dileto assistente, que está sempre me despertando para novas questões. Ao Prof. Guilherme de Alencar Barreto, pelo convívio, conhecimento e orientação, e pela oportunidade de discussões que enriqueceram este trabalho. Aos amigos do laboratório, pela convivência durante todo esse tempo e pelo ótimo ambiente de trabalho. Aos professores e funcionários do Departamento de Engenharia de Teleinformática que, de forma direta ou indireta, participaram do desenvolvimento deste trabalho. À UECE, pelo apoio e pela oportunidade de realizar este trabalho. Aos professores do curso de computação da UECE. Em especial, ao Gustavo, pelas longas conversas sobre redes neurais, agradeço o despertar para esta área e a amizade durante todo este tempo. Ao Celestino, pela amizade e pela torcida durante todo este processo. Ao Thelmo, pelo apoio e pelas discussões filosóficas e matemáticas. Ao Valdísio, pela amizade e por estar sempre pronto a ajudar. Ao Fabiano, pela dedicação, presteza e disponibilidade. Aos amigos da Rua Alegre, pela ajuda no desenvolvimento do pensar. A todos os amigos, pelo apoio, amizade e paciência. iii iv Nesta tese, aborda-se o problema de classificação de dados que estão contaminados com padrões atípicos. Tais padrões, genericamente chamados de outliers, são onipresentes em conjunto de dados multivariados reais, porém sua detecção a priori (i.e antes de treinar um classificador) é uma tarefa de difícil realização. Como conseqüência, uma abordagem reativa, em que se desconfia da presença de outliers somente após um classificador previamente treinado apresentar baixo desempenho, é a mais comum. Várias estratégias podem então ser levadas a cabo a fim de melhorar o desempenho do classificador, dentre elas escolher um classificador mais poderoso computacionalmente ou promover uma limpeza dos dados, eliminando aqueles padrões difíceis de categorizar corretamente. Qualquer que seja a estratégia adotada, a presença de outliers sempre irá requerer maior atenção e cuidado durante o projeto de um classificador de padrões. Tendo estas dificuldades em mente, nesta tese são revisitados conceitos e técnicas provenientes da teoria de regressão robusta, em particular aqueles relacionados à estimação M, adaptando-os ao projeto de classificadores de padrões capazes de lidar automaticamente com outliers. Esta adaptação leva à proposição de versões robustas de dois classificadores de padrões amplamente utilizados na literatura, a saber, o classificador linear dos mínimos quadrados (least squares classifier, LSC) e a máquina de aprendizado extremo (extreme learning machine, ELM). Através de uma ampla gama de experimentos computacionais, usando dados sintéticos e reais, mostra-se que as versões robustas dos classificadores supracitados apresentam desempenho consistentemente superior aos das versões originais.

Tensor-based representations have been widely pursued in the last years due to the increasing number of high-dimensional datasets, which might be better described by the multilinear algebra. In this paper, we introduced a recent pattern recognition technique called Optimum-Path Forest (OPF) in the context of tensor-oriented applications, as well as we evaluated its robustness to space transformations using Multilinear Principal Component Analysis in both face and human action recognition tasks considering image and video datasets. We have shown OPF can obtain more accurate recognition rates in some situations when working on tensor-oriented feature spaces.

Tensor-based representations have been widely pursued in the last years due to the increasing number of high-dimensional datasets, which might be better described by the multilinear algebra. In this paper, we introduced a recent pattern recognition technique called Optimum-Path Forest (OPF) in the context of tensor-oriented applications, as well as we evaluated its robustness to space transformations using Multilinear Principal Component Analysis in both face and human action recognition tasks considering image and video datasets. We have shown OPF can obtain more accurate recognition rates in some situations when working on tensor-oriented feature spaces.

Human motion is the composite consequence of multiple elements—the action performed, an expressive cadence, and a motion signature that captures the distinctive pattern of movement of a particular individual. We develop a new algorithm that is capable of extracting these motion elements and recombining them in novel ways. The algorithm is based on the numerical statistical analysis of motion data spanning multiple subjects performing different types of motions. In particular, we demonstrate that, after our algorithm analyzes a corpus of walking, stair ascending, and stair descending data collected over a group of subjects, it can then observe a sample of walking motion for a new subject and recognize never before seen ascending and descending motions for this new individual. Our approach also yields a generative motion model that can synthesize unseen motions in the distinctive style of this individual. We validate our algorithm using a standard pattern classifier.

In this paper we discuss copositive tensors, which are a natural generalization of the copositive matrices. We present an analysis of some basic properties of copositive tensors; as well as the conditions under which class of copositive tensors and the class of positive semidefinite tensors coincides. Moreover, we have describe several hierarchies that approximates the cone of copositive tensors. The hierarchies are predominantly based on different regimes such as; simplicial partition, rational griding and polynomial conditions. The hierarchies approximates the copositive cone either from inside (inner approximation) or from outside (outer approximation). We will also discuss relationship among different hierarchies.

In this paper we propose two schemes of using the so-called QTTapproximation for the solution of multidimensional parabolic problems. First, we present a simple one-step implicit time integration scheme using an ALS-type solver in the QTT-format. As the second approach, we use the global space-time formulation, resulting in a large block linear system, encapsulating all time steps, and solve it at once in the QTT format. We prove the QTT-rank estimate for certain classes of multivariate potentials and respective solutions in (x, t) variables. The log-linear complexity of storage and the solution time is observed in both spatial and time grid sizes. The method is applied to the Fokker-Planck equation arising from the beadssprings models of polymeric liquids.

This paper proposes using the novelty classifier to face recognition. This classifier is based on novelty filters, proposed by Kohonen. The performance of the new classifier is compared with nearest neighbor classifier, using Euclidian distance. The face data base used for this comparison was the ORL. The face data is extracted using PCA and 2DPCA strategies. Some results are presented: recognition rate versus number of auto vectors, for identification and verification mode and equivalent error rate for verification mode. The results shown that the proposed classifier has a performance better than others previously published, when the leave-one-out method is employed as a test strategy. Best recognition rate of 100% is achieved with this test methodology.  Pattern recognition, face recognition, novelty classifier, principal components analysis. Resumo    Este artigo propõe a utilização do classificado de novidade para reconhecimento de face. Esse classificador é baseado na utilização do filtro de novidade, proposto por Kohonen. O desempenho do novo classificador é comparado com o desempenho do classificador vizinho mais próximo, usando distância euclidiana. A base de dados utilizada para essa comparação foi a base ORL. A informação da face é extraída utilizando PCA e 2DPCA. Os seguintes resultados são apresentados: taxa de reconhecimento versus número de autovetores, no modo de identificação e verificação e taxa de erro equivalente para o modo de verificação. Os resultados obtidos mostraram que o classificador proposto tem um desempenho melhor do que o desempenho do vizinho mais próximo e do que outros classificadores anteriormente publicados usando a mesma base, quando a estratégia Deixa-um-fora (Leave-one-out). A melhor taxa de reconhecimento, 100%, foi obtida com essa metodologia de teste.  Reconhecimento de padrões, reconhecimento facial, filtro de novidades, Análise de Componentes Principais.

This paper studies the concept of fuzzy generalized topologies, which are generalizations of smooth topologies and Chang's fuzzy topologies. A basis of fuzzy generalized topological space will be defined as functions from the family of all fuzzy subsets of a non-empty set X to [0, 1] and some basic properties of their structure will be obtained. Some characterizations of the basis of fuzzy generalized topology, fuzzy generalized cotopology and the product of fuzzy generalized topological spaces will also be shown.

We study two types of problems in this thesis, graph covering problems including the Dominating Set and Edge Cover which are classic combinatorial problems and the Graph Isomorphism Problem with several of its variations. For each of the problems, we provide efficient quadratic unconstrained binary optimization (QUBO) formulations suitable for adiabatic quantum computers, which are viewed as a real-world enhanced model of simulated annealing. The number of qubits (dimension of QUBO matrices) required to solve the graph covering problems are O(n + n lg n) and O(m + n lg n) respectively, where n is the number of vertices and m is the number of edges. We also extend our formulations for the Minimum VertexWeighted Dominating Set problem and Minimum Edge-Weighted Edge Cover problem. For the Graph Isomorphism Problem, we provide two QUBO formulation through two approaches both requiring O(n) variables. We also provide several different formulations for two extensions of the Graph Isomorph...

Abstract: This paper presents the advantages in extending Classical Tensor Algebra (CTA), also known as Kro-necker Algebra, to allow the definition of functions, ie, functional dependencies among its operands. Such ex-tended tensor algebra have been called Generalized ...

The typical formulations of multilinear constraints are presented, and manipulated to reveal their geometric meaning. The inherent geometry of the accepted forms of the multilinears shows that they are a subset of possible constraints. It is shown that the alternative derivation of the trilinear constraint used in the Manual of Photogrammetry 5 th Edition can be easily generalized to alternative forms of the bilinear and quadrilinear constraints. The practical importance of these conclusions is that it allows formation of superior and more easily implemented constraints that are able to avoid degenerative cases. ' , ' j j i i ≠ ≠ .

In this paper, we study a k-uniform directed hypergraph in general form and introduce its adjacency tensor, Laplacian tensor and signless Laplacian tensor. For the k-uniform directed hypergraph H and 0 ≤ α < 1 the convex linear combination of D and A has been defined as Aα = αD+ (1− α)A, where D and A are the degree tensor and the adjacency tensor of H, respectively. We propose some spectral properties of Aα. We also introduce power directed hypergraph and cored directed hypergraph and investigate their α-spectral properties.

The nonnegative inverse eigenvalue problem (NIEP) is: given a family of complex numbers σ = {λ 1 ,. .. , λ n }, find necessary and sufficient conditions for the existence of a nonnegative matrix A of order n with spectrum σ. Loewy and London [R. Loewy, D. London, A note on the inverse eigenvalue problems for nonnegative matrices, Linear and Multilinear Algebra 6 (1978) 83-90] resolved it for n = 3, and for n = 4 when the spectrum is real. In our way of handling the NIEP, we focus our attention on the coefficients of the characteristic polynomial of A. Thus, the NIEP that we consider is: "given k 1 , k 2 ,. .. , k n real numbers, find necessary and sufficient conditions for the existence of a nonnegative matrix A of order n with characteristic polynomial x n + k 1 x n−1 + k 2 x n−2 + • • • + k n ". The coefficients of the characteristic polynomial are closely related to the cyclic structure of the weighted digraph with adjacency matrix A. We introduce a special type of digraph structure, that we shall call EBL, in which this relation is specially simple. We give some results that show the interest of EBL structures. We completely solve the NIEP from

We study the notion of Γ-graded commutative algebra for an arbitrary abelian group Γ. The main examples are the Clifford algebras already treated in . We prove that the Clifford algebras are the only simple finitedimensional associative graded commutative algebras over R or C. Our approach also leads to non-associative graded commutative algebras extending the Clifford algebras.

We show that the two notions of entanglement: the maximum of the geometric measure entanglement and the maximum of the nuclear norm is attained for the same states. We affirm the conjecture of Higuchi-Sudberry on the maximum entangled state of four qubits. We introduce the notion of d-density tensor for mixed d-partite states. We show that d-density tensor is separable if and only if its nuclear norm is 1. We suggest an alternating method for computing the nuclear norm of tensors. We apply the above results to symmetric tensors.

In this paper we consider a geothermal energy storage in which the spatiotemporal temperature distribution is modeled by a heat equation with a convection term. Such storages often are embedded in residential heating systems and control and management require the knowledge of some aggregated characteristics of that temperature distribution in the storage. They describe the input-output behaviour of the storage and the associated energy flows and their response to charging and discharging processes. We aim to derive an efficient approximative description of these characteristics by a low-dimensional system of ODEs. This leads to a model order reduction problem for a large scale linear system of ODEs arising from the semi-discretization of the heat equation combined with a linear algebraic output equation. In a first step we approximate the non time-invariant system of ODEs by a linear time-invariant system. Then we apply Lyapunov balanced truncation model order reduction to approximate the output by a reduced-order system with only a few state equations but almost the same input-output behavior. The paper presents results of extensive numerical experiments showing the efficiency of the applied model order reduction methods. It turns out that only a few suitable chosen ODEs are sufficient to produce good approximations of the input-output behaviour of the storage.

A matrix is called totally nonnegative (TN) if the determinant of every square submatrix is nonnegative and totally positive (TP) if the determinant of every square submatrix is positive. The TP (TN) completion problem asks which partial matrices have a TP (TN) completion. In this paper, a new class of TP- and TN- completable patterns, the border patterns, is identified. This answers an unpublished question about TP-completable patterns that has been outstanding for some time and is the first case of completable patterns with all entries on the border specified. In the process, a new tool is developed: TP line insertion in the second or penultimate line when the first and last entries of the line are specified. Prior results about single unspecified entries are used and generalized

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This paper aims to extend a Krylov subspace technique based on an incomplete orthogonalization of Krylov tensors (as a multidimensional extension of the common Krylov vectors) to solve generalized Sylvester tensor equations via the Einstein product. First, we obtain the tensor form of the quasi-GMRES method, and then we lead to the direct variant of the proposed algorithm. This approach has the great advantage that it uses previous data in each iteration and has a low computational cost. Moreover, an upper bound for the residual norm of the approximate solution is found. Finally, several experimental problems are given to show the acceptable accuracy and efficiency of the presented method.

The equations that define the Lax pairs for generalized principal chiral models can be solved for any constant nondegenerate bilinear form on su(2). The solution is dependent on one free variable that can serve as the spectral parameter. Necessary conditions for the nonconstant metric on SU (2) that define the integrable models are given.

How can we capture the hidden properties from a tensor and a matrix data simultaneously in a fast, accurate, and scalable way? Coupled matrix-tensor factorization (CMTF) is a major tool to extract latent factors from a tensor and matrices at once. Designing an accurate and efficient CMTF method has become more crucial as the size and dimension of real-world data are growing explosively. However, existing methods for CMTF suffer from lack of accuracy, slow running time, and limited scalability. In this paper, we propose S3CMTF, a fast, accurate, and scalable CMTF method. S3CMTF achieves high speed by exploiting the sparsity of real-world tensors, and high accuracy by capturing inter-relations between factors. Also, S3CMTF accomplishes additional speed-up by lock-free parallel SGD update for multi-core shared memory systems. We present two methods, S3CMTF-naive and S3CMTF-opt. S3CMTF-naive is a basic version of S3CMTF, and S3CMTF-opt improves its speed by exploiting intermediate data....

Computer vision plays a crucial role in Advanced Assistance Systems. Most computer vision systems are based on Deep Convolutional Neural Networks (deep CNN) architectures. However, the high computational resource to run a CNN algorithm is demanding. Therefore, the methods to speed up computation have become a relevant research issue. Even though several works on architecture reduction found in the literaturehave not yet been achievedsatisfactory results for embedded real-time system applications. This paper presents an alternative approach based on the Multilinear Feature Space (MFS) method resorting to transfer learning from large CNN architectures. The proposed method uses CNNs to generate feature maps, although it does not work as complexity reduction approach. After the training process, the generated features maps are used to create vector feature space. We use this new vector space to make projections of any new sample to classify them. Our method, named AMFC, uses the transfe...

In this paper, we show that a matrix A in Mn(C) that has n coneigenvectors, where coneigenvalues associated with them are distinct, is condiagonalizable. And also show that if all coneigenvalues of conjugate-normal matrix A be real, then it is symmetric.