Showing papers on "Square matrix published in 2014"

Powers of Tensors and Fast Matrix Multiplication

Abstract: This paper presents a method to analyze the powers of a given trilinear form (a special kind of algebraic constructions also called a tensor) and obtain upper bounds on the asymptotic complexity of matrix multiplication. Compared with existing approaches, this method is based on convex optimization, and thus has polynomial-time complexity. As an application, we use this method to study powers of the construction given by Coppersmith and Winograd [Journal of Symbolic Computation, 1990] and obtain the upper bound $\omega<2.3728639$ on the exponent of square matrix multiplication, which slightly improves the best known upper bound.

Singular value decomposition based low-footprint speaker adaptation and personalization for deep neural network

Abstract: The large number of parameters in deep neural networks (DNN) for automatic speech recognition (ASR) makes speaker adaptation very challenging. It also limits the use of speaker personalization due to the huge storage cost in large-scale deployments. In this paper we address DNN adaptation and personalization issues by presenting two methods based on the singular value decomposition (SVD). The first method uses an SVD to replace the weight matrix of a speaker independent DNN by the product of two low rank matrices. Adaptation is then performed by updating a square matrix inserted between the two low-rank matrices. In the second method, we adapt the full weight matrix but only store the delta matrix - the difference between the original and adapted weight matrices. We decrease the footprint of the adapted model by storing a reduced rank version of the delta matrix via an SVD. The proposed methods were evaluated on short message dictation task. Experimental results show that we can obtain similar accuracy improvements as the previously proposed Kullback-Leibler divergence (KLD) regularized method with far fewer parameters, which only requires 0.89% of the original model storage.

Core–EP inverse

Abstract: In this work, we introduce a notion of ‘core–EP inverse’ for a square matrix which is not essentially of index one. This extends the notion of ‘core inverse’, which was initially defined for the matrices of index one. The properties of matrices having ‘core–EP inverse’ and ‘core–EP generalized inverse’ are studied, and obtained a formula to compute the core–EP generalized inverse from a particular linear combination of minors of given matrix.

Fast matrix completion without the condition number

Abstract: We give the first algorithm for Matrix Completion whose running time and sample complexity is polynomial in the rank of the unknown target matrix, linear in the dimension of the matrix, and logarithmic in the condition number of the matrix. To the best of our knowledge, all previous algorithms either incurred a quadratic dependence on the condition number of the unknown matrix or a quadratic dependence on the dimension of the matrix in the running time. Our algorithm is based on a novel extension of Alternating Minimization which we show has theoretical guarantees under standard assumptions even in the presence of noise.

Lattice-based Key Exchange on Small Integer Solution problem

Abstract: In this paper, we propose a new hard problem, called bilateral inhomogeneous small integer solution (Bi-ISIS), which can be seen as an extension of the small integer solution problem on lattices. The main idea is that, instead of choosing a rectangle matrix, we choose a square matrix with small rank to generate Bi-ISIS problem without affecting the hardness of the underlying SIS problem. Based on this new problem, we present two new hardness problems: computational Bi-ISIS and decisional problems. As a direct application of these problems, we construct a new lattice-based key exchange (KE) protocol, which is analogous to the classic Diffie- Hellman KE protocol. We prove the security of this protocol and show that it provides better security in case of worst-case hardness of lattice problems, relatively efficient implementations, and great simplicity.

Iterative method to solve the generalized coupled Sylvester-transpose linear matrix equations over reflexive or anti-reflexive matrix

Abstract: The iterative method of generalized coupled Sylvester-transpose linear matrix equations A X B + C Y T D = S 1 , E X T F + G Y H = S 2 over reflexive or anti-reflexive matrix pair ( X , Y ) is presented. On the condition that the coupled matrix equations are consistent, we show that the solution pair ( X ∗ , Y ∗ ) proposed by the iterative method can be obtained within finite iterative steps in the absence of roundoff-error for any initial value given a reflexive or anti-reflexive matrix. Moreover, the optimal approximation reflexive or anti-reflexive matrix solution pair to an arbitrary given reflexive or anti-reflexive matrix pair can be derived by searching the least Frobenius norm solution pair of the new generalized coupled Sylvester-transpose linear matrix equations. Finally, some numerical examples are given which illustrate that the introduced iterative algorithm is quite efficient.

The Cubic Simple Matrix Encryption Scheme

Abstract: In this paper, we propose an improved version of the Simple Matrix encryption scheme of PQCrypto2013. The main goal of our construction is to build a system with even stronger security claims. By using square matrices with random quadratic polynomials, we can claim that breaking the system using algebraic attacks is at least as hard as solving a set of random quadratic equations. Furthermore, due to the use of random polynomials in the matrix A, Rank attacks against our scheme are not feasible.

Linear complementarity as absolute value equation solution

Abstract: We consider the linear complementarity problem (LCP): $$Mz+q\ge 0, z\ge 0, z^{\prime }(Mz+q)=0$$ as an absolute value equation (AVE): $$(M+I)z+q=|(M-I)z+q|$$ , where $$M$$ is an $$n\times n$$ square matrix and $$I$$ is the identity matrix. We propose a concave minimization algorithm for solving (AVE) that consists of solving a few linear programs, typically two. The algorithm was tested on 500 consecutively generated random solvable instances of the LCP with $$n=10, 50, 100, 500$$ and 1,000. The algorithm solved $$100\,\%$$ of the test problems to an accuracy of $$10^{-8}$$ by solving 2 or less linear programs per LCP problem.

Rhapsody in fractional

Abstract: This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform

Absolute Value Equation Solution Via Linear Programming

Abstract: By utilizing a dual complementarity property, we propose a new linear programming method for solving the NP-hard absolute value equation (AVE): Ax?|x|=b, where A is an n×n square matrix. The algorithm makes no assumptions on the AVE other than solvability and consists of solving a few linear programs, typically less than four. The algorithm was tested on 500 consecutively generated random solvable instances of the AVE with n=10, 50, 100, 500 and 1000. The algorithm solved 100 % of the test problems to an accuracy of 10?8 by solving an average of 3.3 linear programs per AVE problem.

Nonnegative splittings for rectangular matrices

Abstract: The extension of the nonnegative splitting for rectangular matrices called proper nonnegative splitting is proposed first. Different convergence and comparison theorems for the proper nonnegative splittings are established. The notion of double nonnegative splitting is then generalized to rectangular matrices. Finally, different convergence and comparison results are presented for this decomposition. The case for singular square matrices is also studied.

Eigenvalues of block structured asymmetric random matrices

Abstract: We study the spectrum of an asymmetric random matrix with block structured variances. The rows and columns of the random square matrix are divided into $D$ partitions with arbitrary size (linear in $N$). The parameters of the model are the variances of elements in each block, summarized in $g\in\mathbb{R}^{D\times D}_+$. Using the Hermitization approach and by studying the matrix-valued Stieltjes transform we show that these matrices have a circularly symmetric spectrum, we give an explicit formula for their spectral radius and a set of implicit equations for the full density function. We discuss applications of this model to neural networks.

Diagonal unitary entangling gates and contradiagonal quantum states

Abstract: Nonlocal properties of an ensemble of diagonal random unitary matrices of order $N^2$ are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as $\log N$, in contrast to the $\log N^2$ behavior characteristic to random unitary gates. Entangling power of a diagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantum state $\rho=AA^{\dagger}/N^2$, where the square matrix $A$ is obtained by reshaping the vector of diagonal elements of $U$ of length $N^2$ into a square matrix of order $N$. This fact provides a motivation to study the ensemble of non-hermitian unimodular matrices $A$, with all entries of the same modulus and random phases and the ensemble of quantum states $\rho$, such that all their diagonal entries are equal to $1/N$. Such a state is contradiagonal with respect to the computational basis, in sense that among all unitary equivalent states it maximizes the entropy copied to the environment due to the coarse graining process. The first four moments of the squared singular values of the unimodular ensemble are derived, based on which we conjecture a connection to a recently studied combinatorial object called the "Borel triangle". This allows us to find exactly the mean von Neumann entropy for random phase density matrices and the average entanglement for the corresponding ensemble of bipartite pure states.

On the full exploitation of symmetry in periodic (as well as molecular) self-consistent-field ab initio calculations

Abstract: Use of symmetry can dramatically reduce the computational cost (running time and memory allocation) of self-consistent-field ab initio calculations for molecular and crystalline systems. Crucial for running time is symmetry exploitation in the evaluation of one- and two-electron integrals, diagonalization of the Fock matrix at selected points in reciprocal space, reconstruction of the density matrix. As regards memory allocation, full square matrices (overlap, Fock, and density) in the Atomic Orbital (AO) basis are avoided and a direct transformation from the packed AO to the symmetry adapted crystalline orbital basis is performed, so that the largest matrix to be handled has the size of the largest sub-block in the latter basis. Quantitative examples, referring to the implementation in the CRYSTAL code, are given for high symmetry families of compounds such as carbon fullerenes and nanotubes.

Eigenvectors and minimal bases for some families of Fiedler-like linearizations

Abstract: In this paper, we obtain formulas for the left and right eigenvectors and minimal bases of some families of Fiedler-like linearizations of square matrix polynomials. In particular, for the families of Fiedler pencils, generalized Fiedler pencils and Fiedler pencils with repetition. These formulas allow us to relate the eigenvectors and minimal bases of the linearizations with the ones of the polynomial. Since the eigenvectors appear in the standard formula of the condition number of eigenvalues of matrix polynomials, our results may be used to compare the condition numbers of eigenvalues of the linearizations within these families and the corresponding condition number of the polynomial eigenvalue problem.

An improved Schulz-type iterative method for matrix inversion with application:

Abstract: In this paper, we present an algorithm which could be considered an improvement to the well-known Schulz iteration for finding the inverse of a square matrix iteratively. The convergence of the proposed method is proved and its computational complexity is analysed. The extension of the scheme to generalized outer inverses will be treated. In order to validate the new scheme, we apply it to large sparse matrices alongside the application to preconditioning of practical problems.

The Lin-Bose Problem

Abstract: This brief studies prime factorization problems of a multivariate polynomial matrix. We mainly investigate the Lin-Bose problem proposed by Lin and Bose in the multidimensional system theory context. A simple proof to the Lin-Bose problem is presented in a much easier way to understand. As a by-product, we obtain some properties on modules generated by all the rows of a matrix.

Methodology and Equipments for Analog Circuit Parametric Faults Diagnosis Based on Matrix Eigenvalues

Abstract: A method and corresponding equipments for analog circuit parametric faults diagnosis based on matrix eigenvalues are presented in this paper. The proposed method organizes the discrete samples of response signals of the circuits under test (CUT) into a square matrix and calculates out the maximal and minimal eigenvalues of the square matrix. According to the one-to-one correspondence relationship between the matrix elements and fault cases, fault detection and fault location are achieved by using the maximal and minimal eigenvalues as fault signatures. Two experimental results show that the proposed method has better fault coverage, higher computational efficiency and needs fewer test points than the other state-of-the-art methods. Without the necessary of node-voltage equation or internal structure analysis, solely, depending on the analysis of the output response of CUT to achieve the fault diagnosis, the presented method is particularly suitable for the analog integrated circuit fault diagnosis, and can be extended to solve the fault diagnosis for superconductor digital circuits with finite accessible nodes.

Approximating the Matrix Sign Function Using a Novel IterativeMethod

Abstract: This study presents a matrix iterative method for finding the sign of a square complex matrix. It is shown that the sequence of iterates converges to the sign and has asymptotical stability, provided that the initial matrix is appropriately chosen. Some illustrations are presented to support the theory.

The CP-Matrix Completion Problem

Abstract: A symmetric matrix $C$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $C=BB^T$. The CP-completion problem is to study whether we can assign values to the missing entries of a partial matrix (i.e., a matrix having unknown entries) such that the completed matrix is completely positive. We propose a semidefinite algorithm for solving general CP-completion problems and study its properties. When all of the diagonal entries are given, the algorithm can give a certificate if a partial matrix is not CP-completable, and it almost always gives a CP-completion if it is CP-completable. When diagonal entries are partially given, similar properties hold. Computational experiments are also presented to show how CP-completion problems can be solved.

Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations. Part 2: Kernel separability, space extension, and, series solution via telescopic matrices

Abstract: This work focuses on the Kronecker power series solution of the explicit conical ODEs. This means that the Kronecker power series of the descriptive function vector of the ODEs has only zeroth, first and second Kronecker powers of the unknowns hence the only nonvanishing matrix coefficients are $${\mathbf {F}}_0, {\mathbf {F}}_1$$ and $${\mathbf {F}}_2$$ . We focus on the cases where $${\mathbf {F}}_0$$ also vanishes. These enable us to get and solve a two block term recursive ODE and the accompanying initial conditions. The resulting Kronecker power series’ kernel can be expressed as a binary product whose first factor which in square matrix type and a second factor which is in purely rectangular matrix algebraic structure. The constancy adding space extension separates the temporal behavior of the kernel in a scalar first factor while the second factor is again in rectangular matrix structure. We also show that the definition and use of rectangular eigenvalue problem takes us to constant solution of the original ODEs.