Showing papers on "Circulant matrix published in 2011"

Circulant Preconditioners for Hermitian Toeplitz Systems

Abstract: The solutions of Hermitian positive definite Toeplitz systems $Ax = b$ by the preconditioned conjugate gradient method for three families of circulant preconditioners C is studied. The convergence rates of these iterative methods depend on the spectrum of $C^{ - 1} A$. For a Toeplitz matrix A with entries that are Fourier coefficients of a positive function f in the Wiener class, the invertibility of C is established, as well as that the spectrum of the preconditioned matrix $C^{ - 1} A$ clusters around one. It is proved that if f is $(l + 1)$-times differentiable, with $l > 0$, then the error after $2q$ conjugate gradient steps will decrease like $( (q - 1)! )^{ - 2l} $. It is also shown that if C copies the central diagonals of A, then C minimizes $\| C - A \|_1 $ and $\| C - A \|_\infty $.

Johnson-Lindenstrauss lemma for circulant matrices

Abstract: We prove a variant of a Johnson-Lindenstrauss lemma for matrices with circulant structure. This approach allows to minimize the randomness used, is easy to implement and provides good running times. The price to be paid is the higher dimension of the target space k = O(e−2 log3 n) instead of the classical bound k = O(e−2 log n). © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011 (Supported by DFG Heisenberg (HI 584/3-2); FWF START-Preis Sparse Approximation and Optimization in High Dimensions (Y 432-N15).)

Block circulant matrices and applications in free vibration analysis of cyclically repetitive structures

Abstract: In this paper, block circulant matrices and their properties are investigated. Basic concepts and the necessary theorems are presented and then their applications are discussed. It is shown that a circulant matrix can be considered as the sum of Kronecker products in which the first components have the commutativity property with respect to multiplication. The important fact is that the method for block diagonalization of these matrices is much simpler than the previously developed methods, and one does not need to find an additional matrix for orthogonalization. As it will be shown not only the matrices corresponding to domes in the form of Cartesian product, strong Cartesian product and direct product are circulant, but for other structures such as diamatic domes, pyramid domes, flat double layer grids, and some family of transmission towers these matrices are also block circulant.

On a Family of Circulant Matrices for Quasi-Cyclic Low-Density Generator Matrix Codes

Abstract: We present a new class of sparse and easily invertible circulant matrices that can have a sparse inverse though not being permutation matrices. Their study is useful in the design of quasi-cyclic low-density generator matrix codes that are able to join the inner structure of quasi-cyclic codes with sparse generator matrices, so limiting the number of elementary operations needed for encoding. Circulant matrices of the proposed class permit to hit both targets without resorting to identity or permutation matrices that may penalize the code minimum distance and often cause significant error floors.

Spectral Norm of Circulant-Type Matrices

Abstract: We first discuss the convergence in probability and in distribution of the spectral norm of scaled Toeplitz, circulant, reverse circulant, symmetric circulant, and a class of k-circulant matrices when the input sequence is independent and identically distributed with finite moments of suitable order and the dimension of the matrix tends to ∞. When the input sequence is a stationary two-sided moving average process of infinite order, it is difficult to derive the limiting distribution of the spectral norm, but if the eigenvalues are scaled by the spectral density, then the limits of the maximum of modulus of these scaled eigenvalues can be derived in most of the cases.

On the clique number of integral circulant graphs

Abstract: The concept of gcd-graphs is introduced by Klotz and Sander, which arises as a generalization of unitary Cayley graphs. The gcd-graph $X_n (d_1,...,d_k)$ has vertices $0,1,...,n-1$, and two vertices $x$ and $y$ are adjacent iff $\gcd(x-y,n)\in D = \{d_1,d_2,...,d_k\}$. These graphs are exactly the same as circulant graphs with integral eigenvalues characterized by So. In this paper we deal with the clique number of integral circulant graphs and investigate the conjecture proposed in \cite{klotz07} that clique number divides the number of vertices in the graph $X_n (D)$. We completely solve the problem of finding clique number for integral circulant graphs with exactly one and two divisors. For $k \geqslant 3$, we construct a family of counterexamples and disprove the conjecture in this case.

Further results on the perfect state transfer in integral circulant graphs

Abstract: For a given graph G, denote by A its adjacency matrix and F(t)=exp(iAt). We say that there exist a perfect state transfer (PST) in G if |F(@t)"a"b|=1, for some vertices a,b and a positive real number @t. Such a property is very important for the modeling of quantum spin networks with nearest-neighbor couplings. We consider the existence of the perfect state transfer in integral circulant graphs (circulant graphs with integer eigenvalues). Some results on this topic have already been obtained by Saxena et al. (2007) [5], Basic et al. (2009) [6] and Basic & Petkovic (2009) [7]. In this paper, we show that there exists an integral circulant graph with n vertices having a perfect state transfer if and only if 4|n. Several classes of integral circulant graphs have been found that have a perfect state transfer for the values of n divisible by 4. Moreover, we prove the nonexistence of a PST for several other classes of integral circulant graphs whose order is divisible by 4. These classes cover the class of graphs where the divisor set contains exactly two elements. The obtained results partially answer the main question of which integral circulant graphs have a perfect state transfer.

The energy of integral circulant graphs with prime power order

Abstract: The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. Such a graph can be characterized by its vertex count n and a set D of divisors of n such that its vertex set is Zn and its edge set is {{a,b} : a, b Є Zn; gcd(a-b, n)Є D}. For an integral circulant graph on ps vertices, where p is a prime, we derive a closed formula for its energy in terms of n and D. Moreover, we study minimal and maximal energies for fixed ps and varying divisor sets D.

Stability and stabilizability of special classes of discrete-time positive switched systems

Abstract: In this paper we consider discrete-time positive switched systems, switching among autonomous subsystems, characterized either by cyclic monomial matrices or by circulant matrices. For these two classes of systems, some interesting necessary and sufficient conditions for stability and stabilizability are provided. Such conditions lead to simple algorithms that allow to easily detect whether a given positive switched system is not stabilizable.

New results on the energy of integral circulant graphs

Abstract: Circulant graphs are an important class of interconnection networks in parallel and distributed computing. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer as well. The integral circulant graph $\ICG_n (D)$ has the vertex set $Z_n = \{0, 1, 2, ..., n - 1\}$ and vertices $a$ and $b$ are adjacent if $\gcd(a-b,n)\in D$, where $D \subseteq \{d : d \mid n,\ 1\leq d

Limiting spectral distributions of some band matrices

Abstract: We use the method of moments to establish the limiting spectral distribution (LSD) of appropriately scaled large dimensional random symmetric circulant, reverse circulant, Toeplitz and Hankel matrices which have suitable band structures. The input sequence used to construct these matrices is assumed to be either i.i.d. with mean zero and variance one or independent and appropriate finite fourth moment. The class of LSD includes the normal and the symmetrized square root of chi-square with two degrees of freedom. In several other cases, explicit forms of the limit do not seem to be obtainable but the limits can be shown to be symmetric and their second and the fourth moments can be calculated with some effort. Simulations suggest some further properties of the limits.

On the Covariance Completion Problem Under a Circulant Structure

Abstract: Covariance matrices with a circulant structure arise in the context of discrete-time periodic processes and their significance stems also partly from the fact that they can be diagonalized via a Fourier transformation. This note deals with the problem of completion of partially specified circulant covariance matrices. The particular completion that has maximal determinant (i.e., the so-called maximum entropy completion) was considered in Carli where it was shown that if a single band is unspecified and to be completed, the algebraic restriction that enforces the circulant structure is automatically satisfied and that the inverse of the maximizer has a band of zero values that corresponds to the unspecified band in the data, i.e., it has the Dempster property. The purpose of the present note is to develop an independent proof of this result which in fact extends naturally to any number of missing bands as well as arbitrary missing elements. More specifically, we show that this general fact is a direct consequence of the invariance of the determinant under the group of transformations that leave circulant matrices invariant. A description of the complete set of all positive extensions of partially specified circulant matrices is also given and certain connections between such sets and the factorization of certain polynomials in many variables, facilitated by the circulant structure, is highlighted.

Distance spectra and Distance energy of Integral Circulant Graphs

Abstract: The distance energy of a graph $G$ is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of $G$. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the circulant group, i.e. its adjacency matrix is circulant. A graph is called integral if all eigenvalues of its adjacency matrix are integers. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer. In this paper, we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix $D$. Furthermore, we calculate the distance spectra and distance energy of unitary Cayley graphs. In conclusion, we present two families of pairs $(G_1, G_2)$ of integral circulant graphs with equal distance energy -- in the first family $G_1$ is subgraph of $G_2$, while in the second family the diameter of both graphs is three.

Fast wavelet-based single-particle reconstruction in Cryo-EM

Abstract: This paper presents a novel algorithm for the 3D tomographic inversion problem that arises in single-particle electron cryomicroscopy (Cryo-EM). It is based on two key components: 1) a variational formulation that promotes sparsity in the wavelet domain and 2) the Toeplitz structure of the combined projection/back-projection operator. The first idea has proven to be very effective for the recovery of piecewise-smooth signals, which is confirmed by our numerical experiments. The second idea allows for a computationally efficient implementation of the reconstruction procedure, using only one circulant convolution per iteration.

GGS-groups: order of congruence quotients and Hausdorff dimension

Abstract: If G is a GGS-group defined over a p-adic tree, where p is an odd prime, we calculate the order of the congruence quotients $G_n=G/\Stab_G(n)$ for every n. If G is defined by the vector $e=(e_1,...,e_{p-1})\in\F_p^{p-1}$, the determination of the order of $G_n$ is split into three cases, according as e is non-symmetric, non-constant symmetric, or constant. The formulas that we obtain only depend on p, n, and the rank of the circulant matrix whose first row is e. As a consequence of these formulas, we also obtain the Hausdorff dimension of the closures of all GGS-groups over the p-adic tree.

Diameters of random circulant graphs

Abstract: The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many small-world networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, the worst connected networks are cycles whose diameters increase linearly in the number of nodes. In the present study we consider an intermediate class of examples: Cayley graphs of cyclic groups, also known as circulant graphs or multi-loop networks. We show that the diameter of a random circulant 2k-regular graph with n vertices scales as n^{1/k}, and establish a limit theorem for the distribution of their diameters. We obtain analogous results for the distribution of the average distance and higher moments.

Approximate Model Selection for Large Scale LSSVM

Abstract: Model selection is critical to least squares support vector machine (LSSVM). A major problem of existing model selection approaches of LSSVM is that the inverse of the kernel matrix need to be calculated with O(n 3 ) complexity for each iteration, where n is the number of training examples. It is prohibitive for the large scale application. In this paper, we propose an approximate approach to model selection of LSSVM. We use multilevel circulant matrices to approximate the kernel matrix so that the fast Fourier transform (FFT) can be applied to reduce the computational cost of matrix inverse. With such approximation, we rst design an ecient LSSVM algorithm with O(n log(n)) complexity and theoretically analyze the eect of kernel matrix approximation on the decision function of LSSVM. We further show that the approximate optimal model produced with the multilevel circulant matrix is consistent with the accurate one produced with the original kernel matrix. Under the guarantee of consistency, we present an approximate model selection scheme, whose complexity is signicantly lower than the previous approaches. Experimental results on benchmark datasets demonstrate the eectiveness of approximate model selection.

Computing Steerable Principal Components of a Large Set of Images and Their Rotations

Abstract: We present here an efficient algorithm to compute the Principal Component Analysis (PCA) of a large image set consisting of images and, for each image, the set of its uniform rotations in the plane. We do this by pointing out the block circulant structure of the covariance matrix and utilizing that structure to compute its eigenvectors. We also demonstrate the advantages of this algorithm over similar ones with numerical experiments. Although it is useful in many settings, we illustrate the specific application of the algorithm to the problem of cryo-electron microscopy.