Number Of Nonzero Entries Research Articles

Approximating sparse quadratic programs

Given a matrix A∈Rn×n, we consider the problem of maximizing xTAx subject to the constraint x∈{−1,1}n. This problem, called MaxQP by Charikar and Wirth [FOCS'04], generalizes MaxCut and has natural applications in data clustering and in the study of disordered magnetic phases of matter. Charikar and Wirth showed that the problem admits an Ω(1/lg⁡n) approximation via semidefinite programming, and Alon, Makarychev, Makarychev, and Naor [STOC'05] showed that the same approach yields an Ω(1) approximation when A corresponds to a graph of bounded chromatic number. Both these results rely on solving the semidefinite relaxation of MaxQP, whose currently best running time is O˜(n1.5⋅min⁡{N,n1.5}), where N is the number of nonzero entries in A and O˜ ignores polylogarithmic factors.In this sequel, we abandon the semidefinite approach and design purely combinatorial approximation algorithms for special cases of MaxQP where A is sparse (i.e., has O(n) nonzero entries). Our algorithms are superior to the semidefinite approach in terms of running time, yet are still competitive in terms of their approximation guarantees. More specifically, we show that:•MaxQP admits a (1/2Δ)-approximation in O(nlg⁡n) time, where Δ=O(1) is the maximum degree of the corresponding graph.•Unit MaxQP, where A∈{−1,0,1}n×n, admits a (1/2d)-approximation in O(n) time when the corresponding graph is d-degenerate, and a (1/3δ)-approximation in O(n1.5) time when the corresponding graph has δn edges for δ=O(1).•MaxQP admits a (1−ε)-approximation in O(n) time when the corresponding graph and each of its minors have bounded local treewidth.•Unit MaxQP admits a (1−ε)-approximation in O(n2) time for H-minor free graphs.

Fast Robust Matrix Completion via Entry-Wise l0 -Norm Minimization.

Matrix completion (MC) aims at recovering missing entries, given an incomplete matrix. Existing algorithms for MC are mainly designed for noiseless or Gaussian noise scenarios and, thus, they are not robust to impulsive noise. For outlier resistance, entry-wise lp -norm with and M-estimation are two popular approaches. Yet the optimum selection of p for the entrywise lp -norm-based methods is still an open problem. Besides, M-estimation is limited by a breakdown point, that is, the largest proportion of outliers. In this article, we adopt entrywise l0 -norm, namely, the number of nonzero entries in a matrix, to separate anomalies from the observed matrix. Prior to separation, the Laplacian kernel is exploited for outlier detection, which provides a strategy to automatically update the entrywise l0 -norm penalty parameter. The resultant multivariable optimization problem is addressed by block coordinate descent (BCD), yielding l0 -BCD and l0 -BCD-F. The former detects and separates outliers, as well as its convergence is guaranteed. In contrast, the latter attempts to treat outlier-contaminated elements as missing entries, which leads to higher computational efficiency. Making use of majorization-minimization (MM), we further propose l0 -BCD-MM and l0 -BCD-MM-F for robust non-negative MC where the nonnegativity constraint is handled by a closed-form update. Experimental results of image inpainting and hyperspectral image recovery demonstrate that the suggested algorithms outperform several state-of-the-art methods in terms of recovery accuracy and computational efficiency.

Power-product matrix: nonsingularity, sparsity and determinant

In this paper, we are interested in a special class of integer matrices, namely the power-product matrix, defined with two positive integers n and d. Each matrix element is computed by a power-product of two weak compositions of d into n parts. The power-product matrix has several interesting applications such as the power-sum representation of polynomials and the difference-of-convex-sums-of-squares decomposition of polynomials. We investigate some properties of this matrix including: nonsingularity, sparsity and determinant. Based on techniques in enumerative combinatorics, we prove that the power-product matrix is nonsingular and the number of nonzero entries can be computed exactly. This matrix shows sparse structure which is a good feature in numerical computation of its inverse required in some applications. Special attention is devoted to the computation of the determinant for n = 2 whose explicit formulation is obtained.

Approximating Sparse Quadratic Programs

Given a matrix $A \in \mathbb{R}^{n\times n}$, we consider the problem of maximizing $x^TAx$ subject to the constraint $x \in \{-1,1\}^n$. This problem, called MaxQP by Charikar and Wirth [FOCS'04], generalizes MaxCut and has natural applications in data clustering and in the study of disordered magnetic phases of matter. Charikar and Wirth showed that the problem admits an $\Omega(1/\lg n)$ approximation via semidefinite programming, and Alon, Makarychev, Makarychev, and Naor [STOC'05] showed that the same approach yields an $\Omega(1)$ approximation when $A$ corresponds to a graph of bounded chromatic number. Both these results rely on solving the semidefinite relaxation of MaxQP, whose currently best running time is $\tilde{O}(n^{1.5}\cdot \min\{N,n^{1.5}\})$, where $N$ is the number of nonzero entries in $A$ and $\tilde{O}$ ignores polylogarithmic factors. In this sequel, we abandon the semidefinite approach and design purely combinatorial approximation algorithms for special cases of MaxQP where $A$ is sparse (i.e., has $O(n)$ nonzero entries). Our algorithms are superior to the semidefinite approach in terms of running time, yet are still competitive in terms of their approximation guarantees. More specifically, we show that: - MaxQP admits a $(1/2\Delta)$-approximation in $O(n \lg n)$ time, where $\Delta$ is the maximum degree of the corresponding graph. - UnitMaxQP, where $A \in \{-1,0,1\}^{n\times n}$, admits a $(1/2d)$-approximation in $O(n)$ time when the corresponding graph is $d$-degenerate, and a $(1/3\delta)$-approximation in $O(n^{1.5})$ time when the corresponding graph has $\delta n$ edges. - MaxQP admits a $(1-\varepsilon)$-approximation in $O(n)$ time when the corresponding graph and each of its minors have bounded local treewidth. - UnitMaxQP admits a $(1-\varepsilon)$-approximation in $O(n^2)$ time when the corresponding graph is $H$-minor free.

A fast multiscale Galerkin method for solving a boundary integral equation in a domain with corners

A fast multiscale Galerkin method is proposed for solving the boundary integral equation derived from the Dirichlet problem of the Laplace equation in a domain with corners. It is well known that the integral operator in the equation can be split into two operators, one is noncompact, the other is compact. We design two truncation strategies for the representation matrices of these operators, respectively, which compress these two dense matrices to sparse ones having only 𝒪(2n) number of nonzero entries, where 2n is the number of the wavelet basis functions used in the method. We prove that the proposed truncation strategies do not ruin the stability and convergence rate of the integral equation. Numerical experiments are presented to verify the theoretical results and demonstrate the effectiveness of the method.

Algebraic conditions and the sparsity of spectrally arbitrary patterns

Abstract Given a square matrix A, replacing each of its nonzero entries with the symbol * gives its zero-nonzero pattern. Such a pattern is said to be spectrally arbitrary when it carries essentially no information about the eigenvalues of A. A longstanding open question concerns the smallest possible number of nonzero entries in an n × n spectrally arbitrary pattern. The Generalized 2n Conjecture states that, for a pattern that meets an appropriate irreducibility condition, this number is 2n. An example of Shitov shows that this irreducibility is essential; following his technique, we construct a smaller such example. We then develop an appropriate algebraic condition and apply it computationally to show that, for n ≤ 7, the conjecture does hold for ℝ, and that there are essentially only two possible counterexamples over ℂ. Examining these two patterns, we highlight the problem of determining whether or not either is in fact spectrally arbitrary over ℂ. A general method for making this determination for a pattern remains a major goal; we introduce an algebraic tool that may be helpful.

A Preference-Based Evolutionary Biobjective Approach for Learning Large-Scale Fuzzy Cognitive Maps: An Application to Gene Regulatory Network Reconstruction

Learning large-scale fuzzy cognitive maps (FCMs) with the sparse attribute automatically from time series without prior knowledge remains a challenging problem. Most existing automated learning methods were applied to learn small-scale FCMs, and the learned FCMs are much denser than the maps constructed by human experts. Learning FCMs is the procedure of judging whether there are connecting edges and determining the weights of connecting edges. Thus, we transform the problem of learning FCMs into a biobjective optimization problem with two objects of minimizing the measure error and the number of nonzero entries, respectively. To solve this optimization problem, a preference-based iterative thresholding evolutionary biobjective optimization algorithm for learning FCMs is proposed. The strategy focuses on the knee area of the Pareto front (PF) with preference on the solutions near the true sparsity. Moreover, an initialization operator based on random forest is proposed to increase the speed of convergence toward the PF. The experiments on large-scale synthetic data with varying sizes and densities and the application to the gene regulatory network reconstruction problem have been conducted to demonstrate that our proposal matches or exceeds the existing state-of-the-art FCM learning approaches in most cases in terms of four measures, namely, Data_Error , Out_of_Sample_Error , Model Error, and SS_Mean . The Data_Error obtained by the proposed method can achieve 3.07E-06 even when the number of nodes reaches 200. We also demonstrate the effectiveness of the proposed initialization operator and the preference-based strategy, which can result in a fast convergence speed and higher accuracy.

0–1 matrices whose k-th powers have bounded entries

ABSTRACT Let be the set of 0–1 matrices of order n such that each entry of the k-th powers of these matrices is bounded by t. Let be the maximum number of nonzero entries of a matrix in . Given any positive integer t, we prove that for when n is sufficiently large, and this maximum number is attained at A if and only if A is permutation similar to the upper triangular tournament matrix of order n.

On strong structural controllability and observability of linear time-varying systems: a constructive method

In this paper, we consider the controllability and observability of generalized linear time-varying (LTV) systems whose coefficients are not exactly known. All that is known about these systems is the placement of non-zero entries in their coefficient matrices $(A,B)$. We provide the characterizations in order to judge whether the placements can guarantee the controllability/observability of such LTV systems, regardless of the exact value of each non-zero coefficient. We also present a direct and efficient algorithm with an associated time cost of $O(n+m+\nu)$ to verify the conditions of our characterizations, where $n$ and $m$ denote the number of columns of $A$ and $B$, respectively, and $\nu$ is number of non-zero entries in $(A,B)$.

Success Probability and Area Spectral Efficiency in SCMA Wireless Networks

Sparse code multiple access (SCMA) is emerging as a promising multiple access technology for the fifth-generation wireless networks. By leveraging user-specific SCMA codebooks, SCMA possesses the potential to enable non-orthogonal resource access, and thus leading to the overloading gain. Benefiting from this, SCMA can accommodate more users and achieve a higher throughput than orthogonal frequency division multiple access. In this correspondence, we propose a tractable analytical framework using tools from stochastic geometry to investigate the performance of SCMA wireless networks. In particular, we obtain explicit closed-form expressions for the success probability and the area spectral efficiency (ASE). Based on the derived results, we prove that the ASE is a quasi-concave function of the transmitter density. Moreover, we shed light on the impact of SCMA codeword dimension $K$ and the number of non-zero entries in SCMA codeword $N$ on the network performance. Specifically, we demonstrate that the optimal transmitter density and the maximum ASE scale super-linearly with rate $K^{N}$ . Simulation results validate the accuracy of the presented analysis and exhibit the superiority of SCMA.

Multigrid-Based Inversion for Volumetric Radar Imaging With Asteroid Interior Reconstruction as a Potential Application

This study concentrates on advancing mathematical and computational methodology for radar tomography imaging in which the unknown volumetric velocity distribution of a wave within a bounded domain is to be reconstructed. Our goal is to enable effective simulation and inversion of a large amount of full-wave data within a realistic 2D or 3D geometry. For propagating and inverting the wave, we present a rigorous multigrid-based forward approach which utilizes the finite-difference time-domain method and a nested finite element grid structure. Based on the multigrid approach, we introduce and validate a multiresolution algorithm which allows regularization of the unknown distribution through a coarse-to-fine inversion scheme. In this approach, sparse signals can be effectively inverted, as the coarse fluctuations are reconstructed before the finer ones. Furthermore, the number of nonzero entries in the system matrix can be compressed and thus the inversion procedure can be speeded up. As a test scenario we investigate satellite-based asteroid interior reconstruction. We use both full-wave and projected wave data and estimate the accuracy of the inversion under different error sources: noise and positioning inaccuracies. The results suggest that the present full-wave inversion approach allows recovering the interior with a single satellite recording backscattering data. It seems that robust results can be achieved, when the peak-to-peak signal-to-noise ratio is above 10 dB. Furthermore, it seems that reconstructing the deep interior can be enhanced if two satellites can be utilized in the measurements.

Approximating Nash Equilibria and Dense Subgraphs via an Approximate Version of Carathéodory's Theorem

We present algorithmic applications of an approximate version of Carathéodory's theorem. The theorem states that given a set of vectors $X$ in $\mathbb{R}^d$, for every vector in the convex hull of $X$ there exists an $\varepsilon$-close (under the $p$-norm distance for $2\leq p < \infty$) vector that can be expressed as a convex combination of at most $b$ vectors of $X$, where the bound $b$ depends on $\varepsilon$ and the norm $p$ and is independent of the dimension $d$. This theorem can be derived by instantiating Maurey's lemma, early references to which can be found in the work of Pisier (1981) and Carl (1985). However, in this paper we present a self-contained proof of this result. Using this theorem we establish that in a bimatrix game with $ n \times n$ payoff matrices $A, B$, if the number of nonzero entries in any column of $A+B$ is at most $s$ then an $\varepsilon$-Nash equilibrium of the game can be computed in time $n^{O(\frac{\log s }{\varepsilon^2})}$. This, in particular, gives us a polynomial-time approximation scheme for Nash equilibrium in games with fixed column sparsity $s$. Moreover, for arbitrary bimatrix games the running time of our algorithm matches the best-known upper bound, which was obtained by Lipton, Markakis, and Mehta (2003). The theorem also leads to an additive approximation algorithm for the normalized densest $k$-subgraph problem. Given a graph with $n$ vertices and maximum degree $d$, our algorithm determines a size-$k$ subgraph with normalized density within $\varepsilon$ of the optimal in time $n^{O(\frac{\log d}{\varepsilon^2})}$.

Duality and nonlinear graph Laplacians

In this paper we show that a recent nearly linear time algorithm for solving a system of equations arising from a graph Laplacian can be extended to a large class of nonlinear systems of equations, based on a nonlinear generalization of the graph Laplacian. This result follows from a nonlinear generalization of the notable duality between graph Laplacians and electrical flows. Beyond providing a fast algorithm for a class of nonlinear sets of equations, our work highlights the robustness of spectral graph theory and suggests new directions for research in spectral graph theory.Specifically, we present an iterative algorithm for solving a class of nonlinear system of equations arising from a nonlinear generalization of the graph Laplacian in O˜(k2mlog⁡(kn/ϵ)) iterations, where k is a measure of nonlinearity, n is the number of variables, m is the number of nonzero entries in the generalized graph Laplacian L, ϵ is the solution accuracy and O˜() neglects (non-leading) logarithmic terms. This algorithm is a natural nonlinear extension of the one in Kelner et al. (2013), which solves a linear Laplacian system of equations in nearly linear time. Unlike the linear case, in the nonlinear case each iteration takes O˜(n) time so the total running time is O˜(k2mnlog⁡(kn/ϵ)). For sparse graphs with m=O(n) and fixed k this nonlinear algorithm is O˜(n2log⁡(n/ϵ)) which is slightly faster than standard methods for solving linear equations, which require approximately O(n2.38) time.

Convertible Subspaces of Hessenberg-Type Matrices

We describe subspaces of generalized Hessenberg matrices where the determinant is convertible into the permanent by affixing ± signs. An explicit characterization of convertible Hessenberg-type matrices is presented. We conclude that convertible matrices with the maximum number of nonzero entries can be reduced to a basic set.

On a conjugate directions method for solving strictly convex QP problem

Problem of solving the strictly convex, quadratic programming problem is studied. The idea of conjugate directions is used. First we assume that we know the set of directions conjugate with respect to the hessian of the goal function. We apply n simultaneous directional minimizations along these conjugate directions starting from the same point followed by the addition of the directional corrections. Theorem justifying that the algorithm finds the global minimum of the quadratic goal function is proved. The way of effective construction of the required set of conjugate directions is presented. We start with a vector with zero value entries except the first one. At each step new vector conjugate to the previously generated is constructed whose number of nonzero entries is larger by one than in its predecessor. Conjugate directions obtained by means of the above construction procedure with appropriately selected parameters form an upper triangular matrix which in exact computations is the Cholesky factor of the inverse of the hessian matrix. Computational cost of calculating the inverse factorization is comparable with the cost of the Cholesky factorization of the original second derivative matrix. Calculation of those vectors involves exclusively matrix/vector multiplication and finding an inverse of a diagonal matrix. Some preliminary computational results on some test problems are reported. In the test problems all symmetric, positive definite matrices with dimensions from 14times 14 to 2000times 2000 from the repository of the Florida University were used as the hessians.

Almost Lossless Analog Signal Separation and Probabilistic Uncertainty Relations

We propose an information-theoretic framework for analog signal separation. Specifically, we consider the problem of recovering two analog signals, modeled as general random vectors, from the noiseless sum of linear measurements of the signals. Our framework is inspired by the groundbreaking work of Wu and Verdu (2010) on analog compression and encompasses, inter alia, inpainting, declipping, super-resolution, the recovery of signals corrupted by impulse noise, and the separation of (e.g., audio or video) signals into two distinct components. The main results we report are general achievability bounds for the compression rate, i.e., the number of measurements relative to the dimension of the ambient space the signals live in, under either measurability or Holder continuity imposed on the separator. Furthermore, we find a matching converse for sources of mixed discrete-continuous distribution. For measurable separators our proofs are based on a new probabilistic uncertainty relation, which shows that the intersection of generic subspaces with general sets of sufficiently small Minkowski dimension is empty. Holder continuous separators are dealt with by introducing the concept of regularized probabilistic uncertainty relations. The probabilistic uncertainty relations we develop are inspired by embedding results in dynamical systems theory due to Sauer et al. (1991) and—conceptually—parallel classical Donoho-Stark and Elad-Bruckstein uncertainty principles at the heart of compressed sensing theory. Operationally, the new uncertainty relations take the theory of sparse signal separation beyond traditional sparsity—as measured in terms of the number of non-zero entries—to the more general notion of low description complexity as quantified by Minkowski dimension. Finally, our approach also allows to significantly strengthen key results in Wu and Verdu (2010).

Refined Inertia of Matrix Patterns

This paper explores how the combinatorial arrangement of prescribed zeros in a matrix affects the possible eigenvalues that the matrix can obtain. It demonstrates that there are inertially arbitrary patterns having a digraph with no 2-cycle, unlike what happens for nonzero patterns. A class of patterns is developed that are refined inertially arbitrary but not spectrally arbitrary, making use of the property of a properly signed nest. The paper includes a characterization of the inertially arbitrary and refined inertially arbitrary patterns of order three, as well as the patterns of order four with the least number of nonzero entries.

Symmetric arctic ranks of nonnegative matrices and their linear preservers

A rank one matrix can be factored as for vectors and of appropriate orders. The perimeter of this rank one matrix is the number of nonzero entries in plus the number of nonzero entries in . A matrix of rank k is the sum of k rank one matrices, a rank one decomposition. The perimeter of a matrix A of rank k is the minimum over all rank one of A of the sums of perimeters of the rank one matrices. The arctic rank of a matrix is one half the perimeter. In this article, we characterize the linear operators that preserve the symmetric arctic ranks of nonnegative symmetric matrices.

Exemplification of Dimensional Analysis via MATLAB® Using Eigen Values

In this research article, a set of dimensional physical quantities is transformed into a dimensionless group (or ratio). For a given set of dimensional variables, the physical variables represent the rows and their dimensions represent the columns of a dimensions-matrix. The dimensions-matrix is rearranged both column- and row-wise. The columns are sorted in ascending order based on the column sum and then on the largest negative entry (i.e., cell value). On the other hand, the rows are sorted in descending order based on the number of non-zero entries found in each row and then on the higher first entry. With the aid of MATLAB®, it was found that the proposed method leads to a permutation matrix that has an Eigen vector whose elements represent the exponent for each physical dimensional quantity such that, at the end, a dimensionless group (or ratio) can be formulated, like Schmidt, Nusselt, Reynolds, and Peclet number. The method, however, was found to work well with a set of physical quantities where each is raised to an exponent of ±1.

Restricted Isometry Property on Banded Block Toeplitz Matrices with Application to Multi-Channel Convolutive Source Separation

In compressive sensing (CS), the restricted isometry property (RIP) is an important condition on measurement matrices which guarantees the recovery of sparse signals with undersampled measurements. It has been proved in the prior works that both random (e.g., i.i.d. Gaussian, Bernoulli, $\ldots$ ) and Toeplitz matrices satisfy the RIP with high probability. However, structured matrices, such as banded Toeplitz matrices have drawn more attention since their structures have the advantage of fast matrix multiplication which may decrease the computational complexity of recovery algorithms. In this paper, we show that banded block Toeplitz matrices satisfy the RIP condition with high probability. Banded block Toeplitz matrices can be used in the sparse multi-channel source separation. The banded block Toeplitz matrices decrease the computational complexity while they have fewer number of non-zero entries in comparison to the same dimensional banded Toeplitz matrices. Furthermore, our simulation results show that banded block Toeplitz matrices outperform banded Toeplitz matrices in signal estimation. The analytical RIP bound for banded block Toeplitz matrices is provided in this paper and the RIP bound of sparse Gaussian matrices is also obtained as an upper bound for banded block Toeplitz matrices. Our simulation and analytical results show that sparse Gaussian random matrices do satisfy the RIP condition with high probability. The probability of satisfying the RIP depends on the probability of zero entries.