Showing papers on "Circulant matrix published in 2008"

Graph Clustering Via a Discrete Uncoupling Process

Abstract: A discrete uncoupling process for finite spaces is introduced, called the Markov Cluster Process or the MCL process. The process is the engine for the graph clustering algorithm called the MCL algorithm. The MCL process takes a stochastic matrix as input, and then alternates expansion and inflation, each step defining a stochastic matrix in terms of the previous one. Expansion corresponds with taking the $k$th power of a stochastic matrix, where $k\in\N$. Inflation corresponds with a parametrized operator $\Gamma_r$, $r\geq 0$, that maps the set of (column) stochastic matrices onto itself. The image $\Gamma_r M$ is obtained by raising each entry in $M$ to the $r$th power and rescaling each column to have sum 1 again. In practice the process converges very fast towards a limit that is invariant under both matrix multiplication and inflation, with quadratic convergence around the limit points. The heuristic behind the process is its expected behavior for (Markov) graphs possessing cluster structure. The process is typically applied to the matrix of random walks on a given graph $G$, and the connected components of (the graph associated with) the process limit generically allow a clustering interpretation of $G$. The limit is in general extremely sparse and iterands are sparse in a weighted sense, implying that the MCL algorithm is very fast and highly scalable. Several mathematical properties of the MCL process are established. Most notably, the process (and algorithm) iterands posses structural properties generalizing the mapping from process limits onto clusterings. The inflation operator $\Gamma_r$ maps the class of matrices that are diagonally similar to a symmetric matrix onto itself. The phrase diagonally positive semi-definite (dpsd) is used for matrices that are diagonally similar to a positive semi-definite matrix. For $r\in\N$ and for $M$ a stochastic dpsd matrix, the image $\Gamma_r M$ is again dpsd. Determinantal inequalities satisfied by a dpsd matrix $M$ imply a natural ordering among the diagonal elements of $M$, generalizing the mapping of process limits onto clusterings. The spectrum of $\Gamma_{\infty} M$ is of the form $\{0^{n-k}, 1^k\}$, where $k$ is the number of endclasses of the ordering associated with $M$, and $n$ is the dimension of $M$. This attests to the uncoupling effect of the inflation operator.

Asymptotically Equivalent Sequences of Matrices and Hermitian Block Toeplitz Matrices With Continuous Symbols: Applications to MIMO Systems

Abstract: For the engineering community, Gray's tutorial monograph on Toeplitz and circulant matrices has been, and remains, the best elementary introduction to the Szego theory on large Toeplitz matrices. In this paper, the most important results of the cited monograph are generalized to block Toeplitz (BT) matrices by maintaining the same mathematical tools used by Gray, that is, by using asymptotically equivalent sequences of matrices. As applications of these results, the geometric minimum mean square error (MMSE) for both an infinite-length multivariate linear predictor and an infinite-length decision feedback equalizer (DFE) for multiple-input-multiple-output (MIMO) channels, are obtained as a limit of the corresponding finite-length cases. Similarly, a short derivation of the well-known capacity of a time-invariant MIMO Gaussian channel with intersymbol interference (ISI) and fixed input covariance matrix is also presented.

Image Reconstruction Methods Based on Block Circulant System Matrices

Abstract: An iterative image reconstruction method used with an imaging system that generates projection data, the method comprises: collecting the projection data; choosing a polar or cylindrical image definition comprising a polar or cylindrical grid representation and a number of basis functions positioned according to the polar or cylindrical grid so that the number of basis functions at different radius positions of the polar or cylindrical image grid is a factor of a number of in-plane symmetries between lines of response along which the projection data are measured by the imaging system; obtaining a system probability matrix that relates each of the projection data to each basis function of the polar or cylindrical image definition; restructuring the system probability matrix into a block circulant matrix and converting the system probability matrix in the Fourier domain; storing the projection data into a measurement data vector; providing an initial polar or cylindrical image estimate; for each iteration; recalculating the polar or cylindrical image estimate according to an iterative solver based on forward and back projection operations with the system probability matrix in the Fourier domain; and converting the polar or cylindrical image estimate into a Cartesian image representation to thereby obtain a reconstructed image.

Nonsingularity on Scaled Factor Circulant Matrices

Abstract: In this paper, Mn(F) will denote the set of all n×n matrices over any fieldMatrixs A, R ∈ Mn(F) are called scaled factor circulant matrix and basic scaled factor circulant matrix respectively, where AR = RA We give th ree discriminations by using only the elements in the first row of the scaled factor circulant matrix and the constants d1,d2,,dn in the diagonal matrix D on non singularity Mathematics Subject Classification: 15A21; 65F15

Matrix Convolution Operators on Groups

Abstract: Lebesgue Spaces of Matrix Functions.- Matrix Convolution Operators.- Convolution Semigroups.

Encoder for low-density parity check codes

Abstract: Encoding of a low-density parity check code uses a block-circulant encoding matrix built from circulant matrices. Encoding can include partitioning data into a plurality of data segments. The data segments are each circularly rotated. A plurality of XOR summations are formed for each rotation of the data segments to produce output symbols. The XOR summations use data from the data segments defined by the circulant matrices. Output symbols are produced in parallel for each rotation of the data segments.

Hermitean Cauchy Integral Decomposition of Continuous Functions on Hypersurfaces

Abstract: We consider Holder continuous circulant (2 x 2) matrix functions G(2)(1) defined on the Ahlfors-David regular boundary Gamma of a domain Omega in R-2n. The main goal is to study under which conditions such a function G(2)(1) can be decomposed as G(2)(1) = G(2)(1+) - G(2)(1-), where the components G(2)(1+/-) are extendable to two-sided H-monogenic functions in the interior and the exterior of Omega, respectively. H-monogenicity is a concept from the framework of Hermitean Clifford analysis, a higher dimensional function theory centered around the simultaneous null solutions of two first-order vector-valued differential operators, called Hermitean Dirac operators. H-monogenic functions then are the null solutions of a (2 x 2) matrix Dirac operator, having these Hermitean Dirac operators as its entries; such functions have been crucial for the development of function theoretic results in the Hermitean Clifford context. Copyright (C) 2008 Ricardo Abreu Blaya et al.

Circulant Triangular Fuzzy Number Matrices

Abstract: In this paper, we present some operations on circulant triangular fuzzy numbers matrices (TFNMs). The first row of the circulant matrices play important role in this study. We also study some properties of determinant and adjoint of circulant TFNMs. Finally, we investigate the distance of generalized TFNMs by a systematic process.

Generalized Circulant Densities and a Sufficient Condition for Separability

Abstract: In a series of papers with Kossakowski, the first author has examined properties of densities for which the positive partial transpositrionm (PPT) property can be readily checked. These densities were also investigated from a different perspective by Baumgartner, Hiesmayr and Narnhofer. In this paper we show how the support of such densities can be expressed in terms of lines in a finite geometry and how that same structure lends itself to checking the necessary PPT condition and to a novel sufficient condition for separability.

Unimodular integer circulants

Abstract: We study families of integer circulant matrices and methods for determining which are unimodular. This problem arises in the study of cyclically presented groups, and leads to the following problem concerning polynomials with integer coefficients: given a polynomial f(x) is an element of Z[x], determine all those n is an element of N such that Res(f(x), x(n) - 1) = +/-1. In this paper we describe methods for resolving this problem, including a method based on the use of Strassman's Theorem on p-adic power series, which are effective in many cases. The methods are illustrated with examples arising in the study of cyclically presented groups and further examples which illustrate the strengths and weaknesses of the methods for polynomials of higher degree.

Efficient Encoding Architecture for IEEE 802.16e LDPC Codes

Abstract: The weakness of implementation for LDPC encoder is that conventional binary Matrix Vector Multiplier has many clock cycles which lead to limited throughput. In this letter in order to construct efficient architecture, we target on IEEE 802.16e LDPC encoders. Over the standard H matrices with Circulant Permutation Matrices, we propose semi-parallel architecture by using cyclic right shift registers and exclusive-OR instead of complex Matrix Vector Multipliers. Proposed efficient encoder for IEEE 802.16e LDPC satisfies compact size and high throughput.

All cyclic p-roots of index 3, found by symmetry-preserving calculations

Abstract: When using a Groebner basis to solve the highly symmetric system of algebraic equations defining the cyclic p-roots, one has the feeling that much of the advantage of computerized symbolic algebra over hand calculation is lost through the fact that the symmetry is immediately ``thrown out'' by the calculations. In this paper, the problem of finding (for all relevant primes p) all cyclic p-roots of index 3 is treated with the symmetry preserved through the calculations. Once we had found the relevant formulas, using MAPLE and MATHEMATICA, the calculations could even be made by hand. On the other hand, with respect to a straightforward attack with Groebner basis, it is not even clear how this could be organized for a general p. In other terminologies, our results involve listings of all bi-unimodular sequences constant on the cosets of the group G_0 of cubic residues, or equivalently all circulant complex Hadamard matrices related to G_0. The corresponding problem for bi-unimodular sequences of index 2 was solved by the first named author in 1989 and shortly after solved independently by de la Harpe and Jones in the case p = 1 (mod 4) and by Munemasa and Watatani in the case p = 3 (mod 4).

The travelling salesman problem in symmetric circulant matrices with two stripes

Abstract: The Symmetric Circulant Travelling Salesman Problem asks for the minimum cost tour in a symmetric circulant matrix. The computational complexity of this problem is not known – only upper and lower bounds have been determined. This paper provides a characterisation of the two-stripe case. Instances where the minimum cost of a tour is equal to either the upper or lower bound are recognised. A new construction providing a tour is proposed for the remaining instances, and this leads to a new upper bound that is closer than the previous one.

Mixing of quantum walks on generalized hypercubes

Abstract: We study continuous-time quantum walks on graphs which generalize the hypercube. The only known family of graphs whose quantum walk instantaneously mixes to uniform is the Hamming graphs with small arities. We show that quantum uniform mixing on the hypercube is robust under the addition of perfect matchings but not much else. Our specific results include: • The graph obtained by augmenting the hypercube with an additive matching x ↦ x ⊕ η is instantaneous uniform mixing whenever |η| is even, but with a slower mixing time. This strictly includes the result of Moore and Russell1 on the hypercube. • The class of Hamming graphs H(n,q) is not uniform mixing if and only if q ≥ 5. This is a tight characterization of quantum uniform mixing on Hamming graphs; previously, only the status of H(n,q) with q < 5 was known. • The bunkbed graph whose adjacency matrix is I ⊗ Qn + X ⊗ Af, where Af is a -circulant matrix defined by a Boolean function f, is not uniform mixing if the Fourier transform of f has support of size smaller than 2n-1. This explains why the hypercube is uniform mixing and why the join of two hypercubes is not. Our work exploits the rich spectral structure of the generalized hypercubes and relies heavily on Fourier analysis of group-circulants.

Sparse shift-invariant NMF

Abstract: Non-negative matrix factorization (NMF) has increasingly been used for efficiently decomposing multivariate data into a signal dictionary and corresponding activations. In this paper, we propose an algorithm called sparse shift-invariant NMF (ssiNMF) for learning possibly overcomplete shift- invariant features. This is done by incorporating a circulant property on the features and sparsity constraints on the activations. The circulant property allows us to capture shifts in the features and enables efficient computation by the Fast Fourier Transform. The ssiNMF algorithm turns out to be matrix-free for we need to store only a small number of features. We demonstrate this on a dataset generated from an overcomplete set of bars.

Parallel and Pipelined Architectures for Cyclic Convolution by Block Circulant Formulation Using Low-Complexity Short-Length Algorithms

Abstract: Fully pipelined parallel architectures are derived for high-throughput and reduced-hardware realization of prime-factor cyclic convolution using hardware-efficient modules for short-length rectangular transform (RT). Moreover, a new approach is proposed for the computation of block pseudocyclic convolution using a block cyclic convolution of equal length along with some correction terms, so that the block pseudocyclic representation of cyclic convolution for non-prime-factor-length (N=rP , when r and P are not mutually prime) could be computed efficiently using the algorithms and architectures of short-length cyclic convolutions. Low-complexity algorithms are derived for efficient computation of those error terms, and overall complexities of the proposed technique are estimated for r=2, 3, 4, 6, 8 and 9. The proposed algorithms are used further to design high-throughput and reduced-hardware structures for cyclic convolution where the cofactors are not relatively prime. The proposed structures for high-throughput implementation are found to offer a reduction of nearly 50%-75% of area-delay product over the existing structures for several convolution-lengths. Low-complexity structures for input/output addition units of short length convolutions are derived and used them along with high-throughput modules for hardware-efficient realization of multifactor convolution, which offers nearly 25%-75% reduction of area-delay complexity over the existing structures for various non-prime-factor length convolutions.

Applications of left circulant matrices in signal and image processing

Abstract: We show numerically that the orthogonal transformation corresponding to the eigenvectors of (random) left circulant matrices, has similar performance in signal/image processing like the discrete cosine transform.