Showing papers on "Circulant matrix published in 2021"

Quantum LDPC Codes with Almost Linear Minimum Distance

Abstract: We give a construction of quantum LDPC codes of dimension Θ(log N) and distance Θ(N/log N) as the code length N → ∞. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance Ω(N1-α/2/log N) and dimension Ω(Nα log N), where 0 ≤ α < 1. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixed R < 1 there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least R with, in some sense, optimal circulant size Ω(N/log N) as the code length N → ∞.

Constructing self-dual codes from group rings and reverse circulant matrices

Abstract: In this work, we describe a construction for self-dual codes in which we employ group rings and reverse circulant matrices. By applying the construction directly over different alphabets, and by employing the well known extension and neighbor methods we were able to obtain extremal binary self-dual codes of different lengths of which some have parameters that were not known in the literature before. In particular, we constructed three new codes of length 64, twenty-two new codes of length 68, twelve new codes of length 80 and four new codes of length 92.

Spectral truncations in noncommutative geometry and operator systems

Abstract: In this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces (i.e. imposing an ultraviolet cutoff in momentum space) and with tolerance relations which provide a coarse grain approximation of geometric spaces at a finite resolution. In our new approach the traditional role played by $$C^*$$ -algebras is taken over by operator systems. As part of the techniques we treat $$C^*$$ -envelopes, dual operator systems and stable equivalence. We define a propagation number for operator systems, which we show to be an invariant under stable equivalence and use to compare approximations of the same space. We illustrate our methods for concrete examples obtained by spectral truncations of the circle. These are operator systems of finite-dimensional Toeplitz matrices and their dual operator systems which are given by functions in the group algebra on the integers with support in a fixed interval. It turns out that the cones of positive elements and the pure state spaces for these operator systems possess a very rich structure which we analyze including for the algebraic geometry of the boundary of the positive cone and the metric aspect i.e. the distance on the state space associated to the Dirac operator. The main property of the spectral truncation is that it keeps the isometry group intact. In contrast, if one considers the other finite approximation provided by circulant matrices the isometry group becomes discrete, even though in this case the operator system is a $$C^*$$ -algebra. We analyze this in the context of the finite Fourier transform on the cyclic group. The extension of noncommutative geometry to operator systems allows one to deal with metric spaces up to finite resolution by considering the relation $$d(x,y)< \varepsilon $$ between two points, or more generally a tolerance relation which naturally gives rise to an operator system.

Fast image inpainting strategy based on the space-fractional modified Cahn-Hilliard equations

Abstract: The solution strategy of the space-fractional modified Cahn-Hilliard equation as a tool for the gray value image inpainting model is studied. The existing strategies solve the convexity splitting scheme of the vector-valued Cahn-Hilliard model by Fourier spectral method. In this paper, we constructed a fast solver for the discretized linear systems possessing the saddle-point structure within block-Toeplitz-Toeplitz-block (BTTB) structure arising from the 2D space-fractional modified Cahn-Hilliard equation. The new solver enjoys computational advantage since circulant approximation and fast Fourier transforms (FFTs) can be used for solving the involved linear subsystems. Theoretical analysis shows the spectrum of the preconditioned matrix clusters around 1, which implies the fast convergence rate of the proposed preconditioner. Numerical examples are given to confirm the effectiveness of our method.

Fast implicit difference schemes for time-space fractional diffusion equations with the integral fractional Laplacian

Abstract: In this paper, we develop two fast implicit difference schemes for solving a class of variable-coefficient time–space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the graded L1 formula for the Caputo fractional derivative and a special finite difference discretization for IFL, where the graded mesh can capture the model problem with a weak singularity at initial time. The stability and convergence are rigorously proved via the M-matrix analysis, which is from the spatial discretized matrix of IFL. Moreover, the proposed schemes use the fast sum-of-exponential approximation and Toeplitz matrix algorithms to reduce the computational cost for the nonlocal property of time and space fractional derivatives, respectively. The fast schemes greatly reduce the computational work of solving the discretized linear systems from (Formula presented.) by a direct solver to (Formula presented.) per preconditioned Krylov subspace iteration and a memory requirement from (MN2) to (NNexp), where N and (Nexp ≪) M are the number of spatial and temporal grid nodes, respectively. The spectrum of preconditioned matrix is also given for ensuring the acceleration benefit of circulant preconditioners. Finally, numerical results are presented to show the utility of the proposed methods.

On 2-partition dimension of the circulant graphs

Abstract: The partition dimension is a variant of metric dimension in graphs. It has arising applications in the fields of network designing, robot navigation, pattern recognition and image processing. Let G (V (G) , E (G)) be a connected graph and Γ = {P1, P2, …, Pm} be an ordered m-partition of V (G). The partition representation of vertex v with respect to Γ is an m-vector r (v|Γ) = (d (v, P1) , d (v, P2) , …, d (v, Pm)), where d (v, P) = min {d (v, x) |x ∈ P} is the distance between v and P. If the m-vectors r (v|Γ) differ in at least 2 positions for all v ∈ V (G), then the m-partition is called a 2-partition generator of G. A 2-partition generator of G with minimum cardinality is called a 2-partition basis of G and its cardinality is known as the 2-partition dimension of G. Circulant graphs outperform other network topologies due to their low message delay, high connectivity and survivability, therefore are widely used in telecommunication networks, computer networks, parallel processing systems and social networks. In this paper, we computed partition dimension of circulant graphs Cn (1, 2) for n ≡ 2 (mod 4), n ≥ 18 and hence corrected the result given by Salman et al. [Acta Math. Sin. Engl. Ser. 2012, 28, 1851-1864]. We further computed the 2-partition dimension of Cn (1, 2) for n ≥ 6.

Numerically stable coded matrix computations via circulant and rotation matrix embeddings

Abstract: Polynomial based methods have recently been used in several works for mitigating the effect of stragglers in distributed matrix computations. However, they suffer from serious numerical issues owing to the condition number of the corresponding real Vandermonde-structured recovery matrices. For a system with $n$ worker nodes where $s$ can be stragglers the condition number grows exponentially in n. We present a novel coded computation approach that leverages the properties of circulant permutation and rotation matrices. Our scheme has an optimal recovery threshold and an upper bound on the worst case condition number of our recovery matrices which grows as ≈ $O$ (ns+6); in the practical scenario where $s$ is a constant, this grows polynomially in n. Our schemes leverage the well-behaved conditioning of complex Vandermonde matrices with parameters on the complex unit circle, while still working with computation over the reals. Exhaustive experimental results demonstrate that our proposed method has condition numbers that are orders of magnitude lower than prior work.

Self-dual and LCD double circulant and double negacirculant codes over $\mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q$

Abstract: Let q be an odd prime power, and denote by $${\mathbb {F}}_q$$ the finite field with q elements. In this paper, we consider the ring $$R={\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$ , where $$u^2=u, v^2=v,uv=vu=0$$ and study double circulant and double negacirculant codes over this ring. We first obtain the necessary and sufficient conditions for a double circulant code to be self-dual (resp. LCD). Then we enumerate self-dual and LCD double circulant and double negacirculant codes over R. Last but not the least, we show that the family of Gray images of self-dual and LCD double circulant codes over R are good. Several numerical examples of self-dual and LCD codes over $${\mathbb {F}}_5$$ as the Gray images of these codes over R are given in short lengths.

An All-at-Once Preconditioner for Evolutionary Partial Differential Equations

Abstract: In [McDonald, Pestana, and Wathen, SIAM J. Sci. Comput., 40 (2018), pp. A1012--A1033], a block circulant preconditioner is proposed for all-at-once linear systems arising from evolutionary partial ...

Adaptive Frequency-Domain Normalized Implementations of Widely-Linear Complex-Valued Filter

Abstract: The widely-linear complex-valued least-mean-square (WL-CLMS) algorithm exhibits slow convergence in the presence of non-circular and highly correlated filter input signals. To tackle such an issue with reduced computational complexity, this paper introduces adaptive frequency-domain normalized implementations of widely-linear complex-valued filter. Two normalized algorithms are firstly devised based on the circulant matrices of weight coefficients and the regression vector, respectively. In the design, the normalization matrix using the second-order complementary statistical information of the input signal helps increase the algorithm convergence speed. Then, mean-square and complementary mean-square performance of periodic update frequency-domain widely-linear normalized LMS (P-FDWL-NLMS) algorithm for non-circular complex signals is analyzed. In addition, by introducing a variable-periodic (VP) mechanism, we propose the VP-FDWL-NLMS method that provides faster convergence than the P-FDWL-NLMS scheme. Computer simulation results show the superiority of the proposed approach over the fullband widely-linear complex-valued least-mean-square, augmented affine projection algorithm and its variable step-size version, in terms of both complexity and convergence rate.

On $${\mathbb {A}}$$ A -numerical radius equalities and inequalities for certain operator matrices

Abstract: The main goal of this article is to establish several new $${\mathbb {A}}$$ -numerical radius equalities for $$n\times n$$ circulant, skew circulant, imaginary circulant, imaginary skew circulant, tridiagonal, and anti-tridiagonal operator matrices, where $${\mathbb {A}}$$ is the $$n\times n$$ diagonal operator matrix whose diagonal entries are positive bounded operator A. Some special cases of our results lead to the results of earlier works in the literature, which shows that our results are more general. Further, some pinching type $${\mathbb {A}}$$ -numerical radius inequalities for $$n\times n$$ block operator matrices are given. Some equality conditions are also given. We also provide a concluding section, which may lead to several new problems in this area.

Constructing Independent Spanning Trees on Generalized Recursive Circulant Graphs

Abstract: The generalized recursive circulant networking can be widely used in the design and implementation of interconnection networks. It consists of a series of processors, each is connected through bidirectional, point-to-point communication channels to different neighbors. In this work, we apply the shortest path routing concept to build independent spanning trees on the generalized recursive circulant graphs. The proposed strategy loosen the restricted conditions in previous research and extended the result to a more general vertex setting by design the specific algorithm to deal with the constraint issue.

The lower bound of the network connectivity guaranteeing in-phase synchronization

Abstract: In-phase synchronization is a stable state of identical Kuramoto oscillators coupled on a network with identical positive connections, regardless of network topology. However, this fact does not mean that the networks always synchronize in-phase because other attractors besides the stable state may exist. The critical connectivity μ c is defined as the network connectivity above which only the in-phase state is stable for all the networks. In other words, below μ c, one can find at least one network that has a stable state besides the in-phase sync. The best known evaluation of the value so far is 0.6828 … ≤ μ c ≤ 0.7889. In this paper, focusing on the twisted states of the circulant networks, we provide a method to systematically analyze the linear stability of all possible twisted states on all possible circulant networks. This method using integer programming enables us to find the densest circulant network having a stable twisted state besides the in-phase sync, which breaks a record of the lower bound of the μ c from 0.6828 … to 0.6838 …. We confirm the validity of the theory by numerical simulations of the networks not converging to the in-phase state.

Complexity of the circulant foliation over a graph

Abstract: In the present paper, we investigate the complexity of infinite family of graphs $$H_n=H_n(G_1,\,G_2,\ldots ,G_m)$$ obtained as a circulant foliation over a graph H on m vertices with fibers $$G_1,\,G_2,\ldots ,G_m.$$ Each fiber $$G_i=C_n(s_{i,1},\,s_{i,2},\ldots ,s_{i,k_i})$$ of this foliation is the circulant graph on n vertices with jumps $$s_{i,1},\,s_{i,2},\ldots ,s_{i,k_i}.$$ This family includes the family of generalized Petersen graphs, I-graphs, sandwiches of circulant graphs, discrete torus graphs and others. We obtain a closed formula for the number $$\tau (n)$$ of spanning trees in $$H_n$$ in terms of Chebyshev polynomials, investigate some arithmetical properties of this function and find its asymptotics as $$n\rightarrow \infty .$$

On the reflexive edge strength of the circulant graphs

Abstract: A labeling of a graph is an assignment that carries some sets of graph elements into numbers (usually the non negative integers). The total $ k $-labeling is an assignment $ f_{e} $ from the edge set to the set $ \{1, 2, ..., k_{e} \} $ and assignment $ f_{v} $ from the vertex set to the set $ \{0, 2, 4, ..., 2k_{v} \} $, where $ k = max\{k_{e}, 2k_{v} \} $. An edge irregular reflexive $ k $-labeling of the graph $ G $ is the total $ k $-labeling, if distinct edges have distinct weights, where the edge weight is defined as the sum of label of that edge and the labels of the end vertices. The minimum $ k $ for which the graph $ G $ has an edge irregular reflexive $ k $-labeling is called the reflexive edge strength of the graph $ G $, denoted by $ res(G) $. In this paper we study the edge reflexive irregular $ k $-labeling for some cases of circulant graphs and determine the exact value of the reflexive edge strength for several classes of circulant graphs.

Fast Circulant Tensor Power Method for High-Order Principal Component Analysis

Abstract: To understand high-order intrinsic key patterns in high-dimensional data, tensor decomposition is a more versatile tool for data analysis than standard flat-view matrix models. Several existing tensor models aim to achieve rapid computation of high-order principal components based on the tensor power method. However, since a tensor power method does not enforce orthogonality in subsequently calculated decomposition components, it causes far more challenges on principal component analysis of high-order tensors. To address this problem, several tensor power method variant algorithms incorporating sparsity into decomposition factors have been proposed. However, because these variant algorithms require additional procedures based on data-driven hyper-parameter optimization algorithms, a trade-off between computational cost and convergence exists. In this paper, a novel tensor power method called the fast circulant tensor power method is proposed. The proposed algorithm combines tensor-train decomposition and the power method. Tensor-train decomposition is a high-order tensor decomposition method based on auxiliary unfolding matrix decomposition. Thus, the power method can be embedded into our methodology without any additional processes. Notably, a simple combination of these two methods may cause a local optima problem because the power method only guarantees convergence on each unfolding matrix in tensor-train decomposition. To solve this problem, the circulant updating method is proposed, which globally optimizes all factor vectors by reordering some steps of the factor vector updates. It is experimentally demonstrated that, compared to state-of-the-art tensor power method variant methodologies, the proposed algorithm achieves the lowest computational complexity and quantitatively good performance in various applications including large-scale color image decomposition and convolutional neural network compression.

Total colorings of circulant graphs

Abstract: The total chromatic number χ″(G) is the least number of colors needed to color the vertices and edges of a graph G such that no incident or adjacent elements (vertices or edges) receive the same co...

On 12-regular nut graphs

Abstract: A nut graph is a simple graph whose adjacency matrix is singular with 1-dimensional kernel such that the corresponding eigenvector has no zero entries. In 2020, Fowler et al. characterised for each d ∈ {3, 4, …, 11} all values n such that there exists a d-regular nut graph of order n. In the present paper, we resolve the first open case d = 12, i.e. we show that there exists a 12-regular nut graph of order n if and only if n ≥ 16. We also present a result by which there are infinitely many circulant nut graphs of degree d ≡ 0 (mod 4) and no circulant nut graphs of degree d ≡ 2 (mod 4). The former result partially resolves a question by Fowler et al. on existence of vertex-transitive nut graphs of order n and degree d. We conclude the paper with problems, conjectures and ideas for further work.

A new method for constructing linear codes with small hulls

Abstract: The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull (LCD codes) and one-dimensional hull by employing the positive characteristic analogues of Gauss sums. These codes are quasi-abelian, and sometimes doubly circulant. Some sufficient conditions for a linear code to be an LCD code (resp. a linear code with one-dimensional hull) are presented. It is worth mentioning that we present a lower bound on the minimum distances of the constructed linear codes. As an application, using these conditions, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) with respect to the online Database of Grassl.

Universal Measurement Matrix Design for Sparse and Co-Sparse Signal Recovery

Abstract: Compressed Sensing (CS) avails mutual coherence metric to choose the measurement matrix that is incoherent with dictionary matrix. Random measurement matrices are incoherent with any dictionary, but their highly uncertain elements necessitate large storage and make hardware realization difficult. In this paper deterministic matrices are employed which greatly reduce memory space and computational complexity. To avoid the randomness completely, deterministic sub-sampling is done by choosing rows deterministically rather than randomly, so that matrix can be regenerated during reconstruction without storing it. Also matrices are generated by orthonormalization, which makes them highly incoherent with any dictionary basis. Random matrices like Gaussian, Bernoulli, semi-deterministic matrices like Toeplitz, Circulant and full-deterministic matrices like DFT, DCT, FZC-Circulant are compared. DFT matrix is found to be effective in terms of recovery error and recovery time for all the cases of signal sparsity and is applicable for signals that are sparse in any basis, hence universal.

Fluctuations of linear eigenvalue statistics of reverse circulant and symmetric circulant matrices with independent entries

Abstract: In this article, we study the fluctuations of linear eigenvalue statistics of reverse circulant (RCn) and symmetric circulant (SCn) matrices with independent entries that satisfy some moment conditions. We show that 1nTrϕ(RCn) and 1nTrϕ(SCn) obey the central limit theorem type result, where ϕ is a polynomial test function.