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Showing papers on "Hadamard transform published in 2015"


Journal ArticleDOI
TL;DR: Some inequalities of Hermite-Hadamard type for h-convex functions defined on convex subsets in real or complex linear spaces are given in this article, and applications for norm inequalities are provided as well.
Abstract: Some inequalities of Hermite-Hadamard type for h-convex functions defined on convex subsets in real or complex linear spaces are given. Applications for norm inequalities are provided as well.

100 citations


Journal ArticleDOI
TL;DR: A reasonable concept on the solutions of fractional impulsive Cauchy problems with Hadamard derivative and the corresponding fractional integral equations are established and two fundamental existence results are presented by using standard fixed point methods.

90 citations


Journal ArticleDOI
TL;DR: In this article, an approximate message passing decoder for sparse superposition codes was proposed, whose decoding complexity scales linearly with the size of the design matrix, and it was shown to asymptotically achieve the AWGN capacity with an appropriate power allocation.
Abstract: Sparse superposition codes were recently introduced by Barron and Joseph for reliable communication over the AWGN channel at rates approaching the channel capacity. The codebook is defined in terms of a Gaussian design matrix, and codewords are sparse linear combinations of columns of the matrix. In this paper, we propose an approximate message passing decoder for sparse superposition codes, whose decoding complexity scales linearly with the size of the design matrix. The performance of the decoder is rigorously analyzed and it is shown to asymptotically achieve the AWGN capacity with an appropriate power allocation. Simulation results are provided to demonstrate the performance of the decoder at finite blocklengths. We introduce a power allocation scheme to improve the empirical performance, and demonstrate how the decoding complexity can be significantly reduced by using Hadamard design matrices.

89 citations


Journal ArticleDOI
TL;DR: A new nonsmooth variational model for the restoration of manifold-valued data which includes second order differences in the regularization term is introduced and an algorithm using an inexact cyclic proximal point algorithm is developed.
Abstract: We introduce a new non-smooth variational model for the restoration of manifold-valued data which includes second order differences in the regularization term. While such models were successfully applied for real-valued images, we introduce the second order difference and the corresponding variational models for manifold data, which up to now only existed for cyclic data. The approach requires a combination of techniques from numerical analysis, convex optimization and differential geometry. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machinery of Jacobi fields. For the n-sphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices.

85 citations


Journal ArticleDOI
TL;DR: In this article, the authors theoretically study operations with a four-level superconducting circuit as a two-qubit system and show how to implement iswap gates and Hadamard gates through pulses on transitions between particular pairs of energy levels.
Abstract: We theoretically study operations with a four-level superconducting circuit as a two-qubit system. Using a mapping on a two-qubit system, we show how to implement iswap gates and Hadamard gates through pulses on transitions between particular pairs of energy levels. Our approach allows one to prepare pure two-qubit entangled states with desired form of reduced density matrices of the same purity and, in particular, arbitrary identical reduced states of qubits. We propose using schemes for the Hadamard gate and two-qubit entangled states with identical reduced density matrices in order to verify $logN$ inequalities for Shannon and R\'enyi entropies for the considered noncomposite quantum system.

85 citations


Journal ArticleDOI
Chun-Su Park1
TL;DR: A fast mode decision algorithm for depth intracoding that selectively omits unnecessary depth-modeling modes in the mode decision process with negligible loss of coding efficiency is proposed.
Abstract: The 3D video extension of High Efficiency Video Coding (3D-HEVC) is the state-of-the-art video coding standard for the compression of the multiview video plus depth format. In the 3D-HEVC design, new depth-modeling modes (DMMs) are utilized together with the existing intraprediction modes for depth intracoding. The DMMs can provide more accurate prediction signals and thereby achieve better compression efficiency. However, testing the DMMs in the intramode decision process causes a drastic increase in the computational complexity. In this paper, we propose a fast mode decision algorithm for depth intracoding. The proposed algorithm first performs a simple edge classification in the Hadamard transform domain. Then, based on the edge classification results, the proposed algorithm selectively omits unnecessary DMMs in the mode decision process. Experimental results demonstrate that the proposed algorithm speeds up the mode decision process by up to 37.65% with negligible loss of coding efficiency.

75 citations


Journal ArticleDOI
TL;DR: Simulations suggest that spatial coupling is more robust and allows for better reconstruction at finite code lengths, and it is shown empirically that the use of a fast Hadamard-based operator allows for an efficient reconstruction, both in terms of computational time and memory, and the ability to deal with very large messages.
Abstract: We study the approximate message-passing decoder for sparse superposition coding on the additive white Gaussian noise channel and extend our preliminary work [1]. We use heuristic statistical-physics-based tools such as the cavity and the replica methods for the statistical analysis of the scheme. While superposition codes asymptotically reach the Shannon capacity, we show that our iterative decoder is limited by a phase transition similar to the one that happens in Low Density Parity check codes. We consider two solutions to this problem, that both allow to reach the Shannon capacity: i) a power allocation strategy and ii) the use of spatial coupling, a novelty for these codes that appears to be promising. We present in particular simulations suggesting that spatial coupling is more robust and allows for better reconstruction at finite code lengths. Finally, we show empirically that the use of a fast Hadamard-based operator allows for an efficient reconstruction, both in terms of computational time and memory, and the ability to deal with very large messages.

74 citations


Journal ArticleDOI
TL;DR: In this article, a Hadamard variational formula for p-capacity of convex bodies in R n is established when 1 p n and existence and regularity for 1 p 2.

70 citations


Journal ArticleDOI
TL;DR: In this article, the proximal point method for finding minima of a special class of nonconvex functions on a Hadamard manifold is presented, and it is proved that each accumulation point of this sequence satisfies the necessary optimality conditions.
Abstract: In this article, we present the proximal point method for finding minima of a special class of nonconvex function on a Hadamard manifold. The well definedness of the sequence generated by the proximal point method is established. Moreover, it is proved that each accumulation point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minima is obtained.

62 citations


Journal ArticleDOI
TL;DR: In this article, Hadamard states for the Yang-Mills equation were constructed for the case when the spacetime is ultra-static, but the background solution depends on time.
Abstract: We construct Hadamard states for the Yang–Mills equation linearized around a smooth, space-compact background solution. We assume the spacetime is globally hyperbolic and its Cauchy surface is compact or equal $${\mathbb{R}^d}$$ . We first consider the case when the spacetime is ultra-static, but the background solution depends on time. By methods of pseudodifferential calculus we construct a parametrix for the associated vectorial Klein–Gordon equation. We then obtain Hadamard two-point functions in the gauge theory, acting on Cauchy data. A key role is played by classes of pseudodifferential operators that contain microlocal or spectral type low-energy cutoffs. The general problem is reduced to the ultra-static spacetime case using an extension of the deformation argument of Fulling, Narcowich and Wald. As an aside, we derive a correspondence between Hadamard states and parametrices for the Cauchy problem in ordinary quantum field theory.

54 citations


Journal ArticleDOI
TL;DR: In this paper, the existence and uniqueness of solutions for a fractional integral boundary value problem involving Hadamard type fractional differential equations and integral boundary conditions were studied, based on some classical ideas of fixed point theory.
Abstract: In this paper, we study the existence and uniqueness of solutions for a fractional integral boundary value problem involving Hadamard type fractional differential equations and integral boundary conditions. Our results are new in the present configuration and are based on some classical ideas of fixed point theory. The paper concludes with some illustrative examples.

Journal ArticleDOI
TL;DR: In this article, the behavior of approximate message-passing (AMP), a solver for linear sparse estimation problems such as compressed sensing, when the i.i.d matrices, for which it has been specifically designed, are replaced by structured operators, such as Fourier and Hadamard ones, was investigated.
Abstract: We study the behavior of approximate message-passing (AMP), a solver for linear sparse estimation problems such as compressed sensing, when the i.i.d matrices—for which it has been specifically designed—are replaced by structured operators, such as Fourier and Hadamard ones. We show empirically that after proper randomization, the structure of the operators does not significantly affect the performances of the solver. Furthermore, for some specially designed spatially coupled operators, this allows a computationally fast and memory efficient reconstruction in compressed sensing up to the information-theoretical limit. We also show how this approach can be applied to sparse superposition codes, allowing the AMP decoder to perform at large rates for moderate block length.

Posted Content
TL;DR: A randomized second-order method for optimization known as the Newton Sketch, based on performing an approximate Newton step using a randomly projected or sub-sampled Hessian, is proposed, which has super-linear convergence with exponentially high probability and convergence and complexity guarantees that are independent of condition numbers and related problem-dependent quantities.
Abstract: We propose a randomized second-order method for optimization known as the Newton Sketch: it is based on performing an approximate Newton step using a randomly projected or sub-sampled Hessian. For self-concordant functions, we prove that the algorithm has super-linear convergence with exponentially high probability, with convergence and complexity guarantees that are independent of condition numbers and related problem-dependent quantities. Given a suitable initialization, similar guarantees also hold for strongly convex and smooth objectives without self-concordance. When implemented using randomized projections based on a sub-sampled Hadamard basis, the algorithm typically has substantially lower complexity than Newton's method. We also describe extensions of our methods to programs involving convex constraints that are equipped with self-concordant barriers. We discuss and illustrate applications to linear programs, quadratic programs with convex constraints, logistic regression and other generalized linear models, as well as semidefinite programs.

09 Dec 2015
TL;DR: In this article, the authors defined a new concept so-called tgs-convex function and established some inequalities of Hadamard type via ordinary and Riemann-Liouville integral.
Abstract: In this paper, the Authors de…ned a new concept so-called tgs-convex function and establish some inequalities of Hadamard type via ordinary and Riemann-Liouville integral.

Journal ArticleDOI
TL;DR: Inexact proximal point methods are extended to find singular points for multivalued vector fields on Hadamard manifolds and improve sharply the corresponding results in Li et al. (2009).
Abstract: Inexact proximal point methods are extended to find singular points for multivalued vector fields on Hadamard manifolds. Convergence criteria are established under some mild conditions. In particular, in the case of proximal point algorithm, that is, $$\varepsilon _n=0$$ ? n = 0 for each $$n$$ n , our results improve sharply the corresponding results in Li et al. (2009). Applications to optimization problems, variational inequality problems and gradient methods are also given.

Journal ArticleDOI
TL;DR: In this paper, some Hadamard-type inequalities for functions whose derivatives in absolute values are convex are established and some applications to special means of real numbers are given, and also some applications of their obtained results to get new error bounds for the sum of the midpoint and trapezoidal formulae.

Journal ArticleDOI
TL;DR: The Euclidean cone over a random graph is used as an auxiliary continuous geometric object that allows for the implementation of martingale methods, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graphs.
Abstract: It is shown that there exist a sequence of 3 -regular graphs { G n } n = 1 ∞ and a Hadamard space X such that { G n } n = 1 ∞ forms an expander sequence with respect to X , yet random regular graphs are not expanders with respect to X . This answers a question of the second author and Silberman. The graphs { G n } n = 1 ∞ are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear-time constant-factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods.

Journal ArticleDOI
TL;DR: In this article, a construction of strongly regular Cayley graphs and skew Hadamard difference sets is presented. But the main tools that are employed are index 2 Gauss sums, instead of cyclotomic numbers.
Abstract: In this paper, we give a construction of strongly regular Cayley graphs and a construction of skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes of finite fields, and they generalize the constructions given by Feng and Xiang [10,12]. Three infinite families of strongly regular graphs with new parameters are obtained. The main tools that we employed are index 2 Gauss sums, instead of cyclotomic numbers.

Journal ArticleDOI
TL;DR: A variant of Hadamard (HDM) ratio test is devised to exploit the NC property of the primary signals for spectrum sensing, which is named the NC-HDM algorithm, and it is shown that it is superior to the state-of-the-art algorithms in detection accuracy and/or robustness.
Abstract: Although noncircular (NC) signals are frequently encountered in wireless communications, their statistical property has not yet been utilized in state-of-the-art methods for spectrum sensing. In this paper, a variant of Hadamard (HDM) ratio test is devised to exploit the NC property of the primary signals for spectrum sensing, which is named the NC-HDM algorithm. As the NC-HDM approach is able to exploit full statistical property of the NC signals and handle deviations from independent and identically distributed (IID) noise, it is superior to the state-of-the-art algorithms in detection accuracy and/or robustness. Moreover, performance analysis is conducted for the NC-HDM approach, including the invariant property, false-alarm probability and detection probability. That is, employing the moment-matching Box's Chi-square approximation, the false-alarm probability can be determined. Since the exact moments of the NC-HDM test statistic under the signal-absence hypothesis can be determined and all moments have been matched, the derived false-alarm probability is very accurate, leading to simple and precise computation of the theoretical decision threshold. On the other hand, as the first two exact moments of the NC-HDM test statistic under the signal-presence hypothesis can be precisely calculated, the detection probability based on moment-matching Beta approximation is quite accurate. Numerical results are included to demonstrate the superiority of the NC-HDM approach and validate our theoretical calculations.

Book ChapterDOI
TL;DR: A genetic algorithm GA to search for plateaued boolean functions, which represent suitable candidates for the design of stream ciphers due to their good cryptographic properties, outperforms Clark et al.'s simulated annealing algorithm with respect to the ratio of generated plateaued Boolean functions per number of optimization runs.
Abstract: We propose a genetic algorithm GA to search for plateaued boolean functions, which represent suitable candidates for the design of stream ciphers due to their good cryptographic properties. Using the spectral inversion technique introduced by Clark, Jacob, Maitra and Stanica, our GA encodes the chromosome of a candidate solution as a permutation of a three-valued Walsh spectrum. Additionally, we design specialized crossover and mutation operators so that the swapped positions in the offspring chromosomes correspond to different values in the resulting Walsh spectra. Some tests performed on the set of pseudoboolean functions of $$n=6$$ and $$n=7$$ variables show that in the former case our GA outperforms Clark et al.'s simulated annealing algorithm with respect to the ratio of generated plateaued boolean functions per number of optimization runs.

Journal ArticleDOI
TL;DR: The results for the constructions of explicit bounds and the qualitative properties for the solutions of certain fractional systems with Hadamard derivative with linearization method and de-singular approach are generalized.

Journal ArticleDOI
TL;DR: A new simple optimal repair strategy for (k + m, k) Hadamard MSR codes is proposed, which can considerably reduce the computation compared with the original one during the node repair.
Abstract: The newly presented $(k+m,k)$ Hadamard minimum storage regenerating (MSR) codes are a class of high rate storage codes with optimal repair property for single node failure. In this paper, we propose a new simple optimal repair strategy for $(k+m,k)$ Hadamard MSR codes, which can considerably reduce the computation compared with the original one during the node repair.

Journal ArticleDOI
TL;DR: In this paper, a non-perturbative construction of the fermionic projector in Minkowski space coupled to a time-dependent external potential is given, which is smooth and decays faster than quadratically for large times.
Abstract: We give a non-perturbative construction of the fermionic projector in Minkowski space coupled to a time-dependent external potential which is smooth and decays faster than quadratically for large times. The weak and strong mass oscillation properties are proven. We show that the integral kernel of the fermionic projector is of Hadamard form, provided that the time integral of the spatial sup-norm of the potential satisfies a suitable bound. This gives rise to an algebraic quantum field theory of Dirac fields in an external potential with a distinguished pure quasi-free Hadamard state.

Journal ArticleDOI
TL;DR: In this article, the Hermite-Hadamard type inequal-ities for the class of mappings whose second derivatives at certain powers are s convex in the second sense are established.
Abstract: Some new results related of the left-hand side of the Hermite-Hadamard type inequal- ities for the class of mappings whose second derivatives at certain powers are s convex in the second sense are established. Also, some applications to special means of real numbers are provided. 2010 Mathematics Subject Classification: 26A51; 26D07; 26D10; 26D15

Journal ArticleDOI
TL;DR: An extension of a proximal point algorithm for difference of two convex functions is presented in the context of Riemannian manifolds of nonposite sectional curvature and it is proved that every cluster point is a critical point of the function under consideration.
Abstract: An extension of a proximal point algorithm for difference of two convex functions is presented in the context of Riemannian manifolds of nonposite sectional curvature. If the sequence generated by our algorithm is bounded it is proved that every cluster point is a critical point of the function (not necessarily convex) under consideration, even if minimizations are performed inexactly at each iteration. Application in maximization problems with constraints, within the framework of Hadamard manifolds is presented.

Journal ArticleDOI
TL;DR: In this article, the existence of positive solutions for a class of coupled integral boundary value problems of nonlinear semipositone Hadamard fractional differential equations was studied, and positive solutions were found for a subset of the problems.
Abstract: In this paper, we study the existence of positive solutions for a class of coupled integral boundary value problems of nonlinear semipositone Hadamard fractional differential equations Du(t) + λf(t, u(t), v(t)) = 0, Dv(t) + λg(t, u(t), v(t)) = 0, t ∈ (1, e), λ > 0, u(1) = v(1) = 0, 0 ≤ j ≤ n− 2, u(e) = μ ∫ e 1 v(s) ds s , v(e) = ν ∫ e

Journal ArticleDOI
TL;DR: This work is devoted to accurate detection performance analysis of the Hadamard ratio method for robust spectrum sensing, and derives accurate analytic formulae for detection probability by computing the first and second exact negative moments for the signal-presence hypothesis.
Abstract: Hadamard ratio test is a well-known approach to robust signal detection in multivariate analysis. Recently, it has been exploited for robust spectrum sensing in cognitive radio, but its detection performance is not yet completely analyzed. This work is devoted to accurate detection performance analysis of the Hadamard ratio method for robust spectrum sensing. By computing the first and second exact negative moments for the signal-presence hypothesis along with employing the Beta distribution approximation, we derive accurate analytic formulae for detection probability. This enables us to theoretically evaluate the detection behavior of the Hadamard ratio test. Numerical results are presented to validate our theoretical findings.

Journal ArticleDOI
TL;DR: In this paper, a generalized vector quasi-equilibrium problem (GVQEP) is introduced and studied on Hadamard manifolds and an existence theorem of solutions for the GVQEP is established under some suitable conditions.
Abstract: In this paper, a generalized vector quasi-equilibrium problem (GVQEP) is introduced and studied on Hadamard manifolds. An existence theorem of solutions for the GVQEP is established under some suitable conditions. Some applications to a generalized vector quasi-variational inequality, a generalized vector variational-like inequality and a vector optimization problem are also presented on Hadamard manifolds.

Journal ArticleDOI
TL;DR: In this article, a non-separable correlation between polarization and orbital angular momentum from the same classical vortex beam has been demonstrated experimentally and the Hadamard gates and conditional phase gates have been designed.
Abstract: We perform Bell's measurement for the non-separable correlation between polarization and orbital angular momentum from the same classical vortex beam. The violation of Bell's inequality for such a non-separable classical correlation has been demonstrated experimentally. Based on the classical vortex beam and non-quantum entanglement between the polarization and the orbital angular momentum, the Hadamard gates and conditional phase gates have been designed. Furthermore, a quantum Fourier transform has been implemented experimentally.

Journal ArticleDOI
TL;DR: In this article, a projection-type method for variational inequalities from Euclidean spaces to Hadamard manifolds is proposed, which is well defined whether the solution set of the problem is non-empty or not, under weak assumptions.
Abstract: In this paper, we extend a projection-type method for variational inequalities from Euclidean spaces to Hadamard manifolds. The proposed method has the following nice features: (i) the algorithm is well defined whether the solution set of the problem is non-empty or not, under weak assumptions; (ii) if the solution set is non-empty, then the sequence generated by the method is convergent to the solution, which is closest to the initial point; and (iii) the existence of the solutions to variational inequalities can be verified through the behaviour of the generated sequence. The results presented in this paper generalize and improve some known results given in literatures.