Showing papers on "Circulant matrix published in 2020"

Low-Complexity Linear Equalization for OTFS Modulation

Abstract: In this letter, we propose low-complexity linear equalizers for orthogonal time frequency space (OTFS) modulation that exploit the structure of the effective channel matrix in OTFS. The proposed approach exploits the block circulant nature of the OTFS channel matrix to achieve significant complexity reduction. For an $N\times M$ OTFS system, where $N$ and $M$ are the number of Doppler and delay bins, respectively, the proposed approach gives exact minimum mean square error (MMSE) and zero-forcing (ZF) solutions with just $\mathcal {O}(MN \log MN)$ complexity, while MMSE and ZF solutions using the traditional matrix inversion approach require $\mathcal {O}(M^{3}N^{3})$ complexity. The proposed approach can provide low complexity initial solutions for local search techniques to achieve enhanced bit error performance.

A Fast Block $\alpha$-Circulant Preconditoner for All-at-Once Systems From Wave Equations

Abstract: In this paper, we propose a fast block $\alpha$-circulant preconditioner for solving the nonsymmetric linear system arising from an all-at-once implicit discretization scheme in time for the wave e...

One-bit compressed sensing with partial Gaussian circulant matrices

Abstract: In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy $\delta$, $m\sim \delta^{-4} s\log(N/s\delta)$ measurements suffice to reconstruct the direction of any $s$-sparse vector up to accuracy $\delta$ via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an $\ell_1/\ell_2$ RIP-property. Under a slightly worse dependence on $\delta$, we establish stability with respect to approximate sparsity, as well as full vector recovery results.

Convolutional Phase Retrieval via Gradient Descent

Abstract: We study the convolutional phase retrieval problem, of recovering an unknown signal $x \in \mathbb C^{n} $ from $m$ measurements consisting of the magnitude of its cyclic convolution with a given kernel $a \in \mathbb C^{m} $ . This model is motivated by applications such as channel estimation, optics, and underwater acoustic communication, where the signal of interest is acted on by a given channel/filter, and phase information is difficult or impossible to acquire. We show that when $a$ is random and the number of observations $m$ is sufficiently large, with high probability $x$ can be efficiently recovered up to a global phase shift using a combination of spectral initialization and generalized gradient descent. The main challenge is coping with dependencies in the measurement operator. We overcome this challenge by using ideas from decoupling theory, suprema of chaos processes and the restricted isometry property of random circulant matrices, and recent analysis of alternating minimization methods.

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. The extension of noncommutative geometry to operator systems allows one to deal with metric spaces up to finite resolution by considering the relation $d(x,y)<\epsilon$ between two points, or more generally a tolerance relation which naturally gives rise to an operator system.

Fast conservative numerical algorithm for the coupled fractional Klein-Gordon-Schrödinger equation

Abstract: In this paper, we are concerned with the numerical solutions of the coupled fractional Klein-Gordon-Schrodinger equation. The numerical schemes are constructed by combining the Crank-Nicolson/leap-frog difference methods for the temporal discretization and the Galerkin finite element methods for the spatial discretization. We give a detailed analysis of the conservation properties in the senses of discrete mass and energy. Then the numerical solutions are shown to be unconditionally bounded in L2 −norm, $H^{\frac {\alpha }{2}}-$ semi-norm and $L^{\infty }-$ norm, respectively. Based on the well-known Brouwer fixed-point theorem and the mathematical induction, the unique solvability of the discrete solutions is proved. Moreover, the schemes are proved to be unconditionally convergent with the optimal order $O\left (\tau ^{2}+h^{r+1}\right )$ , where τ is the temporal step, h is the spatial grid size, and r is the order of the selected finite element space. Furthermore, by using the proposed decoupling and iterative algorithms, several numerical examples are included to support theoretical results and show the effectiveness of the schemes. Finally, the fast Krylov subspace solver with suitable circulant preconditioner is designed to effectively solve the Toeplitz-like linear systems. In each iterative step, this method can effectively reduce the memory requirement of above each finite element scheme from ${{O}\left (M^{2}\right )}$ to O(M), and the computational complexity from ${O\left (M^{3}\right )}$ to ${O(M \log M)}$ , where M is the number of grid nodes. Numerical tests are carried out to show that this fast algorithm is more practical than the traditional backslash and LU factorization/Cholesky decomposition methods, in terms of memory requirement and computational cost.

Fast Binary Embeddings and Quantized Compressed Sensing with Structured Matrices

Abstract: This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset $\mathcal{X} \subset \mathbb{R}^n$, associated with the Euclidean metric, with points in the cube $\{\pm 1\}^m$ and we associate the cube with a pseudo-metric that approximates Euclidean distance among points in $\mathcal{X}$. Our methods rely on quantizing fast Johnson-Lindenstrauss embeddings based on bounded orthonormal systems and partial circulant ensembles, both of which admit fast transforms. Our quantization methods utilize noise-shaping, and include Sigma-Delta schemes and distributed noise-shaping schemes. The resulting approximation errors decay polynomially and exponentially fast in $m$, depending on the embedding method. This dramatically outperforms the current decay rates associated with binary embeddings and Hamming distances. Additionally, it is the first such binary embedding result that applies to fast Johnson-Lindenstrauss maps while preserving $\ell_2$ norms. Second, we again consider noise-shaping schemes, albeit this time to quantize compressed sensing measurements arising from bounded orthonormal ensembles and partial circulant matrices. We show that these methods yield a reconstruction error that again decays with the number of measurements (and bits), when using convex optimization for reconstruction. Specifically, for Sigma-Delta schemes, the error decays polynomially in the number of measurements, and it decays exponentially for distributed noise-shaping schemes based on beta encoding. These results are near optimal and the first of their kind dealing with bounded orthonormal systems.

An Efficient Alternating Algorithm for the Lp-Norm Cross-Gradient Joint Inversion of Gravity and Magnetic Data Using the 2-D Fast Fourier Transform

Abstract: An efficient algorithm for the Lp-norm joint inversion of gravity and magnetic data using the cross-gradient constraint is presented. The presented framework incorporates stabilizers that use Lp-norms (0≤ p≤ 2) of the model parameters, and/or the gradient of the model parameters. The formulation is developed from standard approaches for independent inversion of single data sets, and, thus, also facilitates the inclusion of necessary model and data weighting matrices, for example, depth weighting and hard constraint matrices. Using the block Toeplitz Toeplitz block structure of the underlying sensitivity matrices for gravity and magnetic models, when data are obtained on a uniform grid, the blocks for each layer of the depth are embedded in block circulant circulant block matrices. Then, all operations with these matrices are implemented efficiently using 2-D fast Fourier transforms, with a significant reduction in storage requirements. The nonlinear global objective function is minimized iteratively by imposing stationarity on the linear equation that results from applying linearization of the objective function about a starting model. To numerically solve the resulting linear system, at each iteration, the conjugate gradient algorithm is used. This is improved for large scale problems by the introduction of an algorithm in which updates for the magnetic and gravity parameter models are alternated at each iteration, further reducing total computational cost and storage requirements. Numerical results using a complicated 3-D synthetic model and real data sets obtained over the Galinge iron-ore deposit in the Qinghai province, north-west (NW) of China, demonstrate the efficiency of the presented algorithm.

Frequency-Domain Prony Method for Autoregressive Model Identification and Sinusoidal Parameter Estimation

Abstract: In this study, the weighted integral method for identifying differential equation models is extended to a discrete-time system with a difference equation (DE) model and a finite-length sampled data sequence, and obtain a frequency-domain algorithm for short-time signal analysis and frequency estimation. The derivation consists of three steps. 1) Provide the DE (autoregressive model) with unknown coefficients, which is satisfied in a finite observation interval. 2) Discrete Fourier transform (DFT) the DE to obtain algebraic equations (AEs) among the Fourier coefficients. Two mathematical techniques are introduced to maintain the circulant nature of time shifts. 3) Simultaneously solve a sufficient number of AEs with least squares criterion to obtain unknowns exactly when the driving term is absent, or to obtain unknowns that minimize the driving power when it is present. The methods developed enable a decomposed processing of identification and estimation in the frequency domain. Thus, they will be suitable for maximizing statistical efficiency (smallness of estimation error variance), reducing the computational cost, and use in a resolution-enhanced time-frequency analysis of real-world signals. The performance of the proposed methods are compared with those of several DFT-based methods and Cramer–Rao lower bound. Also, the interference effect and its reduction in frequency-decomposed processing are examined.

Low-complexity Linear Equalization for 2×2 MIMO-OTFS Signals

Abstract: Orthogonal time frequency space (OTFS) modulation is a two-dimensional modulation scheme which has superior performance compared to conventional multicarrier modulation schemes. In this paper, we propose low-complexity linear equalizers for a 2 × 2 multiple-input-multiple-output (MIMO) OTFS system. The proposed equalizers are designed by exploiting the structure of the effective delay-Doppler MIMO channel matrix in a MIMO-OTFS system. The channel matrix in a MIMOOTFS system is a block matrix composed of blocks which have a block circulant with circulant block structure. The proposed approach makes use of the properties of block matrices and block circulant matrices to reduce the computational complexity of linear equalizers. For a 2 × 2 MIMO-OTFS system that uses N ×M OTFS modulation, where N and M denote the number of Doppler and delay bins, respectively, the proposed linear equalizers provide exact solution with a computational complexity of $\mathcal{O}$(MN logMN), whereas conventional linear equalizers require a complexity of $\mathcal{O}$(M3N3).

Shortest Path Search Algorithm in Optimal Two-Dimensional Circulant Networks: Implementation for Networks-on-Chip

Abstract: For a family of optimal two-dimensional circulant networks with an analytical description, two new improved versions of the shortest path search algorithm with a constant complexity estimate are obtained. A simple, based on the geometric model of circulant graphs, proof of the formulas used for the shortest path search algorithm is given. Pair exchange algorithms are presented, and their estimates are given for networks-on-chip (NoCs) with a topology in the form of the considered graphs. New versions of the algorithm improve the previously proposed shortest path search algorithm for optimal generalized Petersen graphs with an analytical description. The new proposed algorithm is a promising solution for the use in NoCs which was confirmed by an experimental study while synthesizing NoC communication subsystems and comparing the consumed hardware resources with those when other previously developed routing algorithms.

On self-dual and LCD double circulant and double negacirculant codes over $\mathbb {F}_{q}+u\mathbb {F}_{q}$

Abstract: Double circulant codes of length 2n over the non-local ring $R=\mathbb {F}_{q}+u\mathbb {F}_{q}, u^{2}=u,$ are studied when q is an odd prime power, and − 1 is a square in $\mathbb {F}_{q}$. Double negacirculant codes of length 2n are studied over R when n is even, and q is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length 2n is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length 4n over $\mathbb {F}_{q}$ are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below for n →∞. The parameters of examples of modest lengths are computed. Several such codes are optimal.

Graphs that are cospectral for the distance Laplacian

Abstract: The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $\mathcal{D}^L (G)=T(G)-\mathcal{D} (G)$, where $T(G)$ is the diagonal matrix of row sums of $\mathcal{D}(G)$. Several general methods are established for producing $\mathcal{D}^L$-cospectral graphs that can be used to construct infinite families. Examples are provided to show that various properties are not preserved by $\mathcal{D}^L$-cospectrality, including examples of $\mathcal{D}^L$-cospectral strongly regular and circulant graphs. It is established that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., $|\delta^L_{1}|\geq \cdots \geq |\delta^L_{n}|$, where $\delta^L_{k}$ is the coefficient of $x^k$.

An all-at-once preconditioner for evolutionary partial differential equations.

Abstract: In [McDonald, Pestana and Wathen, \textit{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 differential equations, in which the preconditioned matrix is proven to be diagonalizable and to have identity-plus-low-rank decomposition in the case of the heat equation. In this paper, we generalize the block circulant preconditioner by introducing a small parameter $\epsilon>0$ into the top-right block of the block circulant preconditioner. The implementation of the generalized preconditioner requires the same computational complexity as that of the block circulant one.Theoretically, we prove that (i) the generalization preserves the diagonalizability and the identity-plus-low-rank decomposition; (ii) all eigenvalues of the new preconditioned matrix are clustered at 1 for sufficiently small $\epsilon$; (iii) GMRES method for the preconditioned system has a linear convergence rate independent of size of the linear system when $\epsilon$ is taken to be smaller than or comparable to square root of time-step size. Numerical results are reported to confirm the efficiency of the proposed preconditioner and to show that the generalization improves the performance of block circulant preconditioner.

Fast and exact simulation of Gaussian random fields defined on the sphere cross time

Abstract: We provide a method for fast and exact simulation of Gaussian random fields on the sphere having isotropic covariance functions. The method proposed is then extended to Gaussian random fields defined over the sphere cross time and having covariance functions that depend on geodesic distance in space and on temporal separation. The crux of the method is in the use of block circulant matrices obtained working on regular grids defined over longitude and latitude.

A fast methodology for large-scale focusing inversion of gravity and magnetic data using the structured model matrix and the 2-D fast Fourier transform

Abstract: Focusing inversion of potential field data for the recovery of sparse subsurface structures from surface measurement data on a uniform grid is discussed. For the uniform grid the model sensitivity matrices exhibit block Toeplitz Toeplitz block structure, by blocks for each depth layer of the subsurface. Then, through embedding in circulant matrices, all forward operations with the sensitivity matrix, or its transpose, are realized using the fast two dimensional Fourier transform. Simulations demonstrate that this fast inversion algorithm can be implemented on standard desktop computers with sufficient memory for storage of volumes up to size $n \approx 1M$. The linear systems of equations arising in the focusing inversion algorithm are solved using either Golub Kahan bidiagonalization or randomized singular value decomposition algorithms in which all matrix operations with the sensitivity matrix are implemented using the fast Fourier transform. These two algorithms are contrasted for efficiency for large-scale problems with respect to the sizes of the projected subspaces adopted for the solutions of the linear systems. The presented results confirm earlier studies that the randomized algorithms are to be preferred for the inversion of gravity data, and that it is sufficient to use projected spaces of size approximately $m/8$, for data sets of size $m$. In contrast, the Golub Kahan bidiagonalization leads to more efficient implementations for the inversion of magnetic data sets, and it is again sufficient to use projected spaces of size approximately $m/8$. Moreover, it is sufficient to use projected spaces of size $m/20$ when $m$ is large, $m \approx 50000$, to reconstruct volumes with $n \approx 1M$. Simulations support the presented conclusions and are verified on the inversion of a practical magnetic data set that is obtained over the Wuskwatim Lake region in Manitoba, Canada.

Sketch Kernel Ridge Regression Using Circulant Matrix: Algorithm and Theory

Abstract: Kernel ridge regression (KRR) is a powerful method for nonparametric regression. The time and space complexity of computing the KRR estimate directly are $\mathcal {O}(n^{3})$ and $\mathcal {O}(n^{2})$ , respectively, which are prohibitive for large-scale data sets, where $n$ is the number of data. In this article, we propose a novel random sketch technique based on the circulant matrix that achieves savings in storage space and accelerates the solution of the KRR approximation. The circulant matrix has the following advantages: It can save time complexity by using the fast Fourier transform (FFT) to compute the product of matrix and vector, its space complexity is linear, and the circulant matrix, whose entries in the first column are independent of each other and obey the Gaussian distribution, is almost as effective as the i.i.d. Gaussian random matrix for approximating KRR. Combining the characteristics of the circulant matrix and our careful design, theoretical analysis and experimental results demonstrate that our proposed sketch method, making the estimate kernel methods scalable and practical for large-scale data problems, outperforms the state-of-the-art KRR estimates in time complexity while retaining similar accuracies. Meanwhile, our sketch method provides the theoretical bound that keeps the optimal convergence rate for approximating KRR.

Accelerating the discrete dipole approximation via circulant preconditioning

Abstract: The discrete dipole approximation (DDA) is a popular numerical method for electromagnetic scattering calculations. The standard DDA formulation involves the uniform discretization of the underlying volume integral equation, leading to a linear system of convolution form. This permits a matrix-vector product to be performed with O ( n log n ) complexity via the fast-Fourier transform (FFT). Thus, in principle, the system can be solved rapidly using an iterative method. However, it is well known that the convergence of iterative methods becomes increasing slow as the optical size and refractive index of the scattering obstacle are increased. In this paper, we present a preconditioning strategy based on the multi-level circulant preconditioner of Chan and Olkin [Numer. Algorithms 6, 89 (1994)] and assess its performance for improving this rate of convergence. In particular, we approximate the system matrix by a circulant matrix which can be inverted efficiently using the FFT. We present numerical experiments for scattering by highly oblate non-absorbing hexagonal plates, demonstrating that this approach serves as an effective preconditioning strategy, reducing simulation times by orders of magnitude in many cases. A Matlab implementation of this work is freely available online.

Biased continuous-time random walks with Mittag-Leffler jumps

Abstract: We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

A note on parallel preconditioning for the all-at-once solution of Riesz fractional diffusion equations.

Abstract: The $p$-step backwards difference formula (BDF) for solving the system of ODEs can result in a kind of all-at-once linear systems, which are solved via the parallel-in-time preconditioned Krylov subspace solvers (see McDonald, Pestana, and Wathen [SIAM J. Sci. Comput., 40(2) (2018): A1012-A1033] and Lin and Ng [arXiv:2002.01108, 17 pages]. However, these studies ignored that the $p$-step BDF ($p\geq 2$) is not selfstarting, when they are exploited to solve time-dependent PDEs. In this note, we focus on the 2-step BDF which is often superior to the trapezoidal rule for solving the Riesz fractional diffusion equations, but its resultant all-at-once discretized system is a block triangular Toeplitz system with a low-rank perturbation. Meanwhile, we first give an estimation of the condition number of the all-at-once systems and then adapt the previous work to construct two block circulant (BC) preconditioners. Both the invertibility of these two BC preconditioners and the eigenvalue distributions of preconditioned matrices are discussed in details. The efficient implementation of these BC preconditioners is also presented especially for handling the computation of dense structured Jacobi matrices. Finally, numerical experiments involving both the one- and two-dimensional Riesz fractional diffusion equations are reported to support our theoretical findings.

A Parallel-In-Time Block-Circulant Preconditioner for Optimal Control of Wave Equations

Abstract: In this paper, we propose a new efficient preconditioner for iteratively solving the large-scale indefinite saddle-point sparse linear system, which arises from discretizing the optimality system i...

Exact Recovery of Multichannel Sparse Blind Deconvolution via Gradient Descent

Abstract: We study the multichannel sparse blind deconvolution (MCS-BD) problem, whose task is to simultaneously recover a kernel $a$ and multiple sparse inputs $\{x_i\}_{i=1}^p$ from their circulant convolu...

The tensor t-function: A definition for functions of third-order tensors

Abstract: A definition for functions of multidimensional arrays is presented. The definition is valid for third-order tensors in the tensor t-product formalism, which regards third-order tensors as block circulant matrices. The tensor function definition is shown to have similar properties as standard matrix function definitions in fundamental scenarios. To demonstrate the definition's potential in applications, the notion of network communicability is generalized to third-order tensors and computed for a small-scale example via block Krylov subspace methods for matrix functions. A complexity analysis for these methods in the context of tensors is also provided.