Showing papers on "Circulant matrix published in 2016"

In Defense of Sparse Tracking: Circulant Sparse Tracker

Abstract: Sparse representation has been introduced to visual tracking by finding the best target candidate with minimal reconstruction error within the particle filter framework. However, most sparse representation based trackers have high computational cost, less than promising tracking performance, and limited feature representation. To deal with the above issues, we propose a novel circulant sparse tracker (CST), which exploits circulant target templates. Because of the circulant structure property, CST has the following advantages: (1) It can refine and reduce particles using circular shifts of target templates. (2) The optimization can be efficiently solved entirely in the Fourier domain. (3) High dimensional features can be embedded into CST to significantly improve tracking performance without sacrificing much computation time. Both qualitative and quantitative evaluations on challenging benchmark sequences demonstrate that CST performs better than all other sparse trackers and favorably against state-of-the-art methods.

Learning Support Correlation Filters for Visual Tracking

Abstract: Sampling and budgeting training examples are two essential factors in tracking algorithms based on support vector machines (SVMs) as a trade-off between accuracy and efficiency. Recently, the circulant matrix formed by dense sampling of translated image patches has been utilized in correlation filters for fast tracking. In this paper, we derive an equivalent formulation of a SVM model with circulant matrix expression and present an efficient alternating optimization method for visual tracking. We incorporate the discrete Fourier transform with the proposed alternating optimization process, and pose the tracking problem as an iterative learning of support correlation filters (SCFs) which find the global optimal solution with real-time performance. For a given circulant data matrix with n^2 samples of size n*n, the computational complexity of the proposed algorithm is O(n^2*logn) whereas that of the standard SVM-based approaches is at least O(n^4). In addition, we extend the SCF-based tracking algorithm with multi-channel features, kernel functions, and scale-adaptive approaches to further improve the tracking performance. Experimental results on a large benchmark dataset show that the proposed SCF-based algorithms perform favorably against the state-of-the-art tracking methods in terms of accuracy and speed.

Fast iterative method with a second order implicit difference scheme for time-space fractional convection-diffusion equations

Abstract: In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of time-space fractional convection-diffusion equations (TSFCDEs). To start with, an implicit difference method based on two-sided weighted shifted Grunwald formulae is proposed with a discussion of the stability and convergence. We construct an implicit difference scheme (IDS) and show that it converges with second order accuracy in both time and space. Then, we develop fast solution methods for handling the resulting system of linear equation with the Toeplitz matrix. The fast Krylov subspace solvers with suitable circulant preconditioners are designed to deal with the resulting Toeplitz linear systems. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ and the computational complexity from $O(N^3)$ to $O(N\log N)$ in each iterative step, where $N$ is the number of grid nodes. Extensive numerical example runs show the utility of these methods over the traditional direct solvers of the implicit difference methods, in terms of computational cost and memory requirements.

On Unifying Multi-View Self-Representations for Clustering by Tensor Multi-Rank Minimization

Abstract: In this paper, we address the multi-view subspace clustering problem. Our method utilizes the circulant algebra for tensor, which is constructed by stacking the subspace representation matrices of different views and then rotating, to capture the low rank tensor subspace so that the refinement of the view-specific subspaces can be achieved, as well as the high order correlations underlying multi-view data can be explored.} By introducing a recently proposed tensor factorization, namely tensor-Singular Value Decomposition (t-SVD) \cite{kilmer13}, we can impose a new type of low-rank tensor constraint on the rotated tensor to capture the complementary information from multiple views. Different from traditional unfolding based tensor norm, this low-rank tensor constraint has optimality properties similar to that of matrix rank derived from SVD, so the complementary information among views can be explored more efficiently and thoroughly. The established model, called t-SVD based Multi-view Subspace Clustering (t-SVD-MSC), falls into the applicable scope of augmented Lagrangian method, and its minimization problem can be efficiently solved with theoretical convergence guarantee and relatively low computational complexity. Extensive experimental testing on eight challenging image dataset shows that the proposed method has achieved highly competent objective performance compared to several state-of-the-art multi-view clustering methods.

On the Construction of Lightweight Circulant Involutory MDS Matrices

Abstract: In the present paper, we investigate the problem of constructing MDS matrices with as few bit XOR operations as possible. The key contribution of the present paper is constructing MDS matrices with entries in the set of $$m\times m$$ non-singular matrices over $$\mathbb {F}_2$$ directly, and the linear transformations we used to construct MDS matrices are not assumed pairwise commutative. With this method, it is shown that circulant involutory MDS matrices, which have been proved do not exist over the finite field $$\mathbb {F}_{2^m}$$, can be constructed by using non-commutative entries. Some constructions of $$4\times 4$$ and $$5\times 5$$ circulant involutory MDS matrices are given when $$m=4,8$$. To the best of our knowledge, it is the first time that circulant involutory MDS matrices have been constructed. Furthermore, some lower bounds on XORs that required to evaluate one row of circulant and Hadamard MDS matrices of order 4 are given when $$m=4,8$$. Some constructions achieving the bound are also given, which have fewer XORs than previous constructions.

Adaptive Control and Synchronization of Halvorsen Circulant Chaotic Systems

Abstract: In this research work, we describe Halvorsen circulant chaotic systems and its qualitative properties. We show that Halvorsen circulant chaotic system is dissipative and that it has an unstable equilibrium at the origin. The Lyapunov exponents of Halvorsen circulant chaotic system are obtained as \(L_1 =0.8109\), \(L_2 = 0\) and \(L_3 = -4.6255\). The Kaplan-Yorke dimension of the Halvorsen circulant chaotic system is obtained as \(D_{KY} = 2.1753\). Next, this work describes the adaptive control of the Halvorsen circulant chaotic system with unknown parameters. Also, this work describes the adaptive synchronization of the identical Halvorsen circulant chaotic systems with unknown parameters. The adaptive feedback control and synchronization results are proved using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results for the Halvorsen circulant chaotic system.

An eigenvalue problem for even order tensors with its applications

Abstract: In this paper, we study an eigenvalue problem for even order tensors. Using the matrix unfolding of even order tensors, we can establish the relationship between a tensor eigenvalue problem and a multilevel matrix eigenvalue problem. By considering a higher order singular value decomposition of a tensor, we show that higher order singular values are the square root of the eigenvalues of the product of the tensor and its conjugate transpose. This result is similar to that in matrix case. Also we study an eigenvalue problem for Toeplitz/circulant tensors, and give the lower and upper bounds of eigenvalues of Toeplitz tensors. An application in image restoration is also discussed.

Some Invariants of Circulant Graphs

Abstract: Topological indices and polynomials are predicting properties like boiling points, fracture toughness, heat of formation, etc., of different materials, and thus save us from extra experimental burden. In this article we compute many topological indices for the family of circulant graphs. At first, we give a general closed form of M-polynomial of this family and recover many degree-based topological indices out of it. We also compute Zagreb indices and Zagreb polynomials of this family. Our results extend many existing results.

Tensor CP Decomposition With Structured Factor Matrices: Algorithms and Performance

Abstract: The canonical polyadic decomposition (CPD) of high-order tensors, also known as Candecomp/Parafac, is very useful for representing and analyzing multidimensional data. This paper considers a CPD model having structured matrix factors, as e.g. Toeplitz, Hankel or circulant matrices, and studies its associated estimation problem. This model arises in signal processing applications such as Wiener-Hammerstein system identification and cumulant-based wireless communication channel estimation. After introducing a general formulation of the considered structured CPD (SCPD), we derive closed-form expressions for the Cramer-Rao bound (CRB) of its parameters under the presence of additive white Gaussian noise. Formulas for special cases of interest, as when the CPD contains identical factors, are also provided. Aiming at a more relevant statistical evaluation from a practical standpoint, we discuss the application of our formulas in a Bayesian context, where prior distributions are assigned to the model parameters. Three existing algorithms for computing SCPDs are then described: a constrained alternating least squares (CALS) algorithm, a subspace-based solution and an algebraic solution for SCPDs with circulant factors. Subsequently, we present three numerical simulation scenarios, in which several specialized estimators based on these algorithms are proposed for concrete examples of SCPD involving circulant factors. In particular, the third scenario concerns the identification of a Wiener-Hammerstein system via the SCPD of an associated Volterra kernel. The statistical performance of the proposed estimators is assessed via Monte Carlo simulations, by comparing their Bayesian mean-square error with the expected CRB.

Perfect state transfer in Laplacian quantum walk

Abstract: For a graph G and a related symmetric matrix M, the continuous-time quantum walk on G relative to M is defined as the unitary matrix $$U(t) = \exp (-itM)$$U(t)=exp(-itM), where t varies over the reals. Perfect state transfer occurs between vertices u and v at time $$\tau $$? if the (u, v)-entry of $$U(\tau )$$U(?) has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer. If an n-vertex graph has perfect state transfer at time $$\tau $$? relative to the Laplacian, then so does its complement if $$n\tau \in 2\pi {\mathbb {Z}}$$n??2?Z. As a corollary, the join of $$\overline{K}_{2}$$K¯2 with any m-vertex graph has perfect state transfer relative to the Laplacian if and only if $$m \equiv 2\pmod {4}$$m?2(mod4). This was previously known for the join of $$\overline{K}_{2}$$K¯2 with a clique (Bose et al. in Int J Quant Inf 7:713---723, 2009). If a graph G has perfect state transfer at time $$\tau $$? relative to the normalized Laplacian, then so does the weak product $$G \times H$$G×H if for any normalized Laplacian eigenvalues $$\lambda $$? of G and $$\mu $$μ of H, we have $$\mu (\lambda -1)\tau \in 2\pi {\mathbb {Z}}$$μ(?-1)??2?Z. As a corollary, a weak product of $$P_{3}$$P3 with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of $$P_{3}$$P3 has perfect state transfer relative to the adjacency matrix. As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (Godsil in Discret Math 312(1):129---147, 2011).

Fast Fourier Orthogonalization

Abstract: The classical fast Fourier transform (FFT) allows to compute in quasi-linear time the product of two polynomials, in the circular convolution ring R[x]/(xd -1) --- a task that naively requires quadratic time. Equivalently, it allows to accelerate matrix-vector products when the matrix is circulant. In this work, we discover that the ideas of the FFT can be applied to speed up the orthogonalization process of matrices with circulant blocks of size d x d. We show that, when d is composite, it is possible to proceed to the orthogonalization in an inductive way ---up to an appropriate re-indexation of rows and columns. This leads to a structured Gram-Schmidt decomposition. In turn, this structured Gram-Schmidt decomposition accelerates a cornerstone lattice algorithm: the nearest plane algorithm. The complexity of both algorithms may be brought down to Θ(d log d).Our results easily extend to cyclotomic rings, and can be adapted to Gaussian samplers. This finds applications in lattice-based cryptography, improving the performances of trapdoor functions.

Circulant tensors with applications to spectral hypergraph theory and stochastic process

Abstract: Circulant tensors naturally arise from stochastic process and spectral hypergraph theory. The joint moments of stochastic processes are symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of circulant hypergraphs are also symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of directed circulant hypergraphs are circulant tensors, but they are not symmetric in general. In this paper, we study spectral properties of circulant tensors and their applications in spectral hypergraph theory and stochastic process. We show that in certain cases, the largest H-eigenvalue of a circulant tensor can be explicitly identified. In particular, the largest H-eigenvalue of a nonnegative circulant tensor can be explicitly identified. This confirms the results in circulant hypergraphs and directed circulant hypergraphs. We prove that an even order circulant B$_0$ tensor is always positive semi-definite. This shows that the Laplacian tensor and the signless Laplacian tensor of a directed circulant even-uniform hypergraph are positive semi-definite. If a stochastic process is $m$th order stationary, where $m$ is even, then its $m$th order moment, which is a circulant tensor, must be positive semi-definite. In this paper, we give various conditions for an even order circulant tensor to be positive semi-definite.

Lightweight MDS Generalized Circulant Matrices

Abstract: In this article, we analyze the circulant structure of generalized circulant matrices to reduce the search space for finding lightweight MDS matrices. We first show that the implementation of circulant matrices can be serialized and can achieve similar area requirement and clock cycle performance as a serial-based implementation. By proving many new properties and equivalence classes for circulant matrices, we greatly reduce the search space for finding lightweight maximum distance separable MDS circulant matrices. We also generalize the circulant structure and propose a new class of matrices, called cyclic matrices, which preserve the benefits of circulant matrices and, in addition, have the potential of being self-invertible. In this new class of matrices, we obtain not only the MDS matrices with the least XOR gates requirement for dimensions from $$3 \times 3$$ to $$8 \times 8$$ in $${\text {GF}}2^4$$ and $${\text {GF}}2^8$$, but also involutory MDS matrices which was proven to be non-existence in the class of circulant matrices. To the best of our knowledge, the latter matrices are the first of its kind, which have a similar matrix structure as circulant matrices and are involutory and MDS simultaneously. Compared to the existing best known lightweight matrices, our new candidates either outperform or match them in terms of XOR gates required for a hardware implementation. Notably, our work is generic and independent of the metric for lightweight. Hence, our work is applicable for improving the search for efficient circulant matrices under other metrics besides XORi¾?gates.

Recycling Randomness with Structure for Sublinear time Kernel Expansions

Abstract: We propose a scheme for recycling Gaussian random vectors into structured matrices to approximate various kernel functions in sublinear time via random embeddings. Our framework includes the Fastfood construction as a special case, but also extends to Circulant, Toeplitz and Hankel matrices, and the broader family of structured matrices that are characterized by the concept of low-displacement rank. We introduce notions of coherence and graph-theoretic structural constants that control the approximation quality, and prove unbiasedness and low-variance properties of random feature maps that arise within our framework. For the case of low-displacement matrices, we show how the degree of structure and randomness can be controlled to reduce statistical variance at the cost of increased computation and storage requirements. Empirical results strongly support our theory and justify the use of a broader family of structured matrices for scaling up kernel methods using random features.

Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

Abstract: High-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant pre-conditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(N logN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

Bayesian compressive sensing with circulant matrix for spectrum sensing in cognitive radio networks

Abstract: For wideband spectrum sensing, compressive sensing has been proposed as a solution to speed up the high dimensional signals sensing and reduce the computational complexity. Compressive sensing consists of acquiring the essential information from a sparse signal and recovering it at the receiver based on an efficient sampling matrix and a reconstruction technique. In order to deal with the uncertainty, improve the signal acquisition performance, and reduce the randomness during the sensing and reconstruction processes, compressive sensing requires a robust sampling matrix and an efficient reconstruction technique. In this paper, we propose an approach that combines the advantages of a Circulant matrix with Bayesian models. This approach is implemented, extensively tested, and using several metrics its results are compared to those of £1 norm minimization with Circulant and random matrices. These metrics are Mean Square Error, reconstruction error, correlation, recovery time, sampling time, and processing time. The results show that our technique is faster and more efficient in compressing and recovering signals.

Independence Complexes of Well-Covered Circulant Graphs

Abstract: We study the independence complexes of families of well-covered circulant graphs discovered by Boros–Gurvich–Milanic, Brown–Hoshino, and Moussi. Because these graphs are well-covered, their independence complexes are pure simplicial complexes. We determine when these pure complexes have extra combinatorial (e.g., vertex decomposable, shellable) or topological (e.g., Cohen–Macaulay, Buchsbaum) properties. We also provide a table of all well-covered circulant graphs on 16 or less vertices, and for each such graph, determine if it is vertex decomposable, shellable, Cohen–Macaulay, and/or Buchsbaum. A highlight of this search is an example of a graph whose independence complex is shellable but not vertex decomposable.

Group Symmetric Robust Covariance Estimation

Abstract: In this paper, we consider Tyler’s robust covariance ${\rm M}$ -estimator under group symmetry constraints. We assume that the covariance matrix is invariant to the conjugation action of a unitary matrix group, referred to as group symmetry. Examples of group symmetric structures include circulant, perHermitian, and proper quaternion matrices. We introduce a group symmetric version of Tyler’s estimator (STyler) and provide an iterative fixed point algorithm to compute it. The classical results claim that at least $n=p+1$ sample points in general position are necessary to ensure the existence and uniqueness of Tyler’s estimator, where $p$ is the ambient dimension. We show that the STyler requires significantly less samples. In some groups, even two samples are enough to guarantee its existence and uniqueness. In addition, in the case of elliptical populations, we provide high probability bounds on the error of the STyler. These, too, quantify the advantage of exploiting the symmetry structure. Finally, these theoretical results are supported by numerical simulations.

Optimized Design of Finite-Length Separable Circulant-Based Spatially-Coupled Codes: An Absorbing Set-Based Analysis

Abstract: In this paper, we characterize the finite-length performance of separable circulant-based spatially-coupled (SCB-SC) LDPC codes for transmission over the additive white Gaussian noise channel. For a general class of finite-length graph-based codes, it is known that the existence of small absorbing sets causes a performance degradation in the error floor regime. We first present the mathematical conditions for the existence of absorbing sets in binary SCB-SC codes. This analysis enables us to find the exact number of absorbing sets as a function of the design parameters. In particular, our results show that the choice of the cutting vector affects the number of absorbing sets and, therefore, the error floor performance of the code. For a fixed column weight, we find provably optimal cutting vectors that result in the least number of absorbing sets. Furthermore, we extend our analysis to nonbinary SCB-SC codes, where we show that the choice of the cutting vector is not as critical as in the binary case. We provide an algorithm which provably removes the problematic nonbinary absorbing sets from nonbinary SCB-SC codes by informed selection of edge labels. Our simulation results show the superior error floor performance of our designed binary and nonbinary SCB-SC codes compared with binary unstructured and nonbinary quasi-cyclic SC codes available in the open literature.

Recycling randomness with structure for sublinear time kernel expansions

Abstract: We propose a scheme for recycling Gaussian random vectors into structured matrices to approximate various kernel functions in sublinear time via random embeddings. Our framework includes the Fastfood construction of Le et al. (2013) as a special case, but also extends to Circulant, Toeplitz and Hankel matrices, and the broader family of structured matrices that are characterized by the concept of low-displacement rank. We introduce notions of coherence and graph-theoretic structural constants that control the approximation quality, and prove unbiasedness and low-variance properties of random feature maps that arise within our framework. For the case of low-displacement matrices, we show how the degree of structure and randomness can be controlled to reduce statistical variance at the cost of increased computation and storage requirements. Empirical results strongly support our theory and justify the use of a broader family of structured matrices for scaling up kernel methods using random features.

On k -circulant matrices (with geometric sequence)

Abstract: Let k be a nonzero complex number. In this paper we show how the inverse of a nonsingular k-circulant matrix can be obtained. The method is used to determine the inverse of a nonsingular k-circulant matrix with geometric sequence. If k = 1, then we get the result presented in the paper A.C.F. Bueno, Right Circulant Matrices With Geometric Progression, Int. J. Appl. Math. Res. 1(4) (2012), 593– 603. Also, we derive the Moore-Penrose inverse of a singular k-circulant matrix with geometric sequence. At the end of the paper, we illustrate the obtained results by examples.

Kansa-RBF Algorithms for Elliptic Problems in Axisymmetric Domains

Abstract: We employ a Kansa-radial basis function method for the numerical solution of elliptic boundary value problems in three-dimensional axisymmetric domains. We consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy--Navier equations. Appropriate discretizations lead to linear systems, the coefficient matrices of which possess block circulant structures. These systems are then solved efficiently by means of matrix decomposition algorithms and fast Fourier transforms. Methods for choosing appropriate values of the shape parameter are also proposed. The effectiveness of the proposed algorithms is demonstrated by considering several numerical examples.

On the bounds for the spectral norms of geometric circulant matrices

Abstract: In this paper, we define a geometric circulant matrix whose entries are the generalized Fibonacci numbers and hyperharmonic Fibonacci numbers. Then we give upper and lower bounds for the spectral norms of these matrices.

On self-dual double negacirculant codes

Abstract: Double negacirculant (DN) codes are the analogues in odd characteristic of double circulant codes. Self-dual DN codes of odd dimension are shown to be consta-dihedral. Exact counting formulae are derived for DN codes. The special class of length a power of two is studied by means of Dickson polynomials, and is shown to contain families of codes with relative distances satisfying a modified Gilbert-Varshamov bound.

On some properties of Toeplitz matrices

Abstract: In this paper, we investigate some properties of Toeplitz matrices with respect to different matrix products. We also give some results regarding circulant matrices, skew-circulant matrices and approximation by Toeplitz matrices over the field of complex numbers.