Nonzero Entries Research Articles

A remark on ranks of sign patterns

For any δ>0, we construct a real matrix A with at most rank(A)−δ zeros in every row and every column, such that any matrix obtained from A by replacing every nonzero entry with a rational number of the same sign has rank greater than rank(A).

A trace bound for integer-diagonal positive semidefinite matrices

AbstractWe prove that an n-by-n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.

Minimum fill-in: Inapproximability and almost tight lower bounds

Given an n×n sparse symmetric matrix with m nonzero entries, performing Gaussian elimination may turn some zeroes into nonzero values, so called fill-ins. The minimum fill-in problem asks whether it is possible to perform the elimination with at most k fill-ins. We exclude the existence of polynomial time approximation schemes for this problem, assuming P≠NP, and the existence of 2O(n1−δ)-time approximation schemes for any positive δ, assuming the Exponential Time Hypothesis. We also give a 2O(k1/2−δ)⋅nO(1) parameterized lower bound. All these results come as corollaries of a new reduction from vertex cover to the minimum fill-in problem, which might be of its own interest: All previous reductions for similar problems start from some kind of graph layout problems, and hence have limited use in understanding their fine-grained complexity.

A group lasso based sparse KNN classifier

Sparse features have been shown effective in applications of computer vision, machine learning, signal processing, etc. Group sparsity was proposed by considering that there exist natural group structures in many problems. Previous research mainly focuses on improving the way to extract sparse features, such as lasso, group lasso, overlapped group lasso, sparse group lasso. In existing work, sparse features are usually taken as input for classifiers, such as SVM, KNN, or SRC (Sparse Representation based Classification). In this paper, we find that, instead of using sparse group features as input for classifiers, sparse group features are good candidates for selection of most relevant classes/groups. We design a new classifier to improve classification accuracy: (1) we use sparse group lasso to select K most relevant classes/groups, which makes this approach robust, because it filters out unrelated classes/groups in group level, instead of individual sample level; (2) KSVD is used to get exact desired sparsity (k nonzero entries) and thus eliminates the difficulty of hyperparameter tuning; (3) simple summation of regression weights within each class/group contains sufficient class discriminant information, and the chance of a sample belonging to a specific class is denoted simply by the summation of corresponding regression weights within each class, which is in line with the need of Explainable AI (XAI). The K most relevant groups/classes can be considered as K neighbors of the correct class. Thus, we call this classifier Group Lasso based Sparse KNN (GLSKNN). Compared to 8 other approaches, GLSKNN classifier outperforms other methods in term of classification accuracy for two public image datasets and images with different occlusion/noise levels.

Signal-Dependent Performance Analysis of Orthogonal Matching Pursuit for Exact Sparse Recovery

Exact recovery of $K$-sparse signals $x \in \mathbb{R}^{n}$ from linear measurements $y=Ax$, where $A\in \mathbb{R}^{m\times n}$ is a sensing matrix, arises from many applications. The orthogonal matching pursuit (OMP) algorithm is widely used for reconstructing $x$. A fundamental question in the performance analysis of OMP is the characterizations of the probability of exact recovery of $x$ for random matrix $A$ and the minimal $m$ to guarantee a target recovery performance. In many practical applications, in addition to sparsity, $x$ also has some additional properties. This paper shows that these properties can be used to refine the answer to the above question. In this paper, we first show that the prior information of the nonzero entries of $x$ can be used to provide an upper bound on $\|x\|_1^2/\|x\|_2^2$. Then, we use this upper bound to develop a lower bound on the probability of exact recovery of $x$ using OMP in $K$ iterations. Furthermore, we develop a lower bound on the number of measurements $m$ to guarantee that the exact recovery probability using $K$ iterations of OMP is no smaller than a given target probability. Finally, we show that when $K=O(\sqrt{\ln n})$, as both $n$ and $K$ go to infinity, for any $0<\zeta\leq 1/\sqrt{\pi}$, $m=2K\ln (n/\zeta)$ measurements are sufficient to ensure that the probability of exact recovering any $K$-sparse $x$ is no lower than $1-\zeta$ with $K$ iterations of OMP. For $K$-sparse $\alpha$-strongly decaying signals and for $K$-sparse $x$ whose nonzero entries independently and identically follow the Gaussian distribution, the number of measurements sufficient for exact recovery with probability no lower than $1-\zeta$ reduces further to $m=(\sqrt{K}+4\sqrt{\frac{\alpha+1}{\alpha-1}\ln(n/\zeta)})^2$ and asymptotically $m\approx 1.9K\ln (n/\zeta)$, respectively.

A Linear-Time Algorithm for Generalized Trust Region Subproblems

In this paper, we provide the first provable linear-time (in terms of the number of nonzero entries of the input) algorithm for approximately solving the generalized trust region subproblem (GTRS) of minimizing a quadratic function over a quadratic constraint under some regularity condition. Our algorithm is motivated by and extends a recent linear-time algorithm for the trust region subproblem by Hazan and Koren [Math. Program., 158 (2016), pp. 363--381]. However, due to the nonconvexity and noncompactness of the feasible region, such an extension is nontrivial. Our main contribution is to demonstrate that under some regularity condition, the optimal solution is in a compact and convex set and lower and upper bounds of the optimal value can be computed in linear time. Using these properties, we develop a linear-time algorithm for the GTRS.

PhaseEqual: Convex Phase Retrieval via Alternating Direction Method of Multipliers

The problem of estimating an unknown complex signal from its magnitude-only measurements is a classical problem known as phase retrieval. This problem arises naturally because in some applications, it is difficult or costly to measure the phase of the measurements. In this paper, we propose a new convex formulation, referred to as PhaseEqual, for solving the phase retrieval problem. Similar to PhaseMax T. Goldstein and C. Studer, 2018 by Goldstein and Studer (IEE TIT, 2018), PhaseEqual works in the natural signal space and thus is computationally efficient. We then extend PhaseEqual to the sparse signal case, with the proposed convex formulation termed as compressed PhaseEqual. Different from existing convex compressed phase retrieval methods, the proposed compressed PhaseEqual formulation does not involve any regularization parameter and thus is free of the parameter tuning issue which is always tricky in practice. Order-wise recovery conditions for PhaseEqual and its sparse version (i.e. compressed PhaseEqual) are analyzed. Our theoretical results show that PhaseEqual (resp. compressed PhaseEqual) achieves perfect recovery with $\mathcal {O}(n)$ (resp. $\mathcal {O}(k\text{log}\frac{n}{k})$ ) magnitude measurements, provided that a well-correlated reference vector is available, where $k$ and $n$ denote the number of nonzero entries in the complex sparse signal and the dimension of the signal, respectively. Simulation results are provided to illustrate the effectiveness of the proposed methods.

Near-Linear Time Algorithm for $n$-Fold ILPs via Color Coding

We study an important case of integer linear programs (ILPs) of the form $\max\{c^Tx \ \vert\ \mathcal Ax = b, l \leq x \leq u,\, x \in \mathbb{Z}^{n t} \} $ with $n t$ variables and lower and upper bounds $\ell, u\in\mathbb Z^{nt}$. In $n$-fold ILPs nonzero entries only appear in the first $r$ rows of the matrix $\mathcal A$ and in small blocks of size $s\times t$ along the diagonal underneath. Despite this restriction, many optimization problems can be expressed in this form. It is known that $n$-fold ILPs are fixed-parameter tractable (FPT) regarding the parameters $s, r,$ and $\Delta$, where $\Delta$ is the greatest absolute value of any entry in $\mathcal A$. The state-of-the-art technique is a local search algorithm that subsequently moves in an improving direction where the number of iterations and the search for such an improving direction each take time $\Omega(n)$. This leads to a running time quadratic in $n$. We introduce a technique based on color coding which allows us to compute these improving directions in logarithmic time after a single initialization step. This yields an algorithm for $n$-fold ILPs with a running time that is near-linear in $nt$, the number of variables. More precisely, our algorithm runs in time $(rs\Delta)^{\mathcal{O}(r^2s + s^2)} L^2 nt \log^{\mathcal{O}(1)}(nt)$, where $L$ is the encoding length of the largest integer in the input. Further, in contrast to the algorithms in recent literature, we do not need to solve the LP relaxation in order to handle unbounded variables. Instead we give a structural lemma to introduce appropriate bounds. On the other hand, if we are given such an LP solution, the running time can be decreased by a factor of $L$.

Initial State Causes the Structural Balance of Complex Networks With Dynamical Models

What initial conditions may lead the dynamic complex network into structural balance? In the sense of social network, the solution to this problem explains that the initial connection relations between individuals are one of the reasons to achieve social stability. In order to obtain the solution, the special Riccati differential equation is chosen as the mathematic model of the dynamical complex network in this paper. Different from the existing results, we focus on the exact solution to the Riccati differential equation and draw the mathematic initial conditions from the exact solution. By using the existing results about the real logarithm of the matrix, the unique exact solution is obtained under the given initial conditions. This solution is a real continuous and differentiable symmetric matrix, and it is seen from which that how the initial state can lead to approximate asymptotically the structural balance more clearly. It turns out that the plus or minus sign of eigenvalues of the initial link matrix affects the asymptotic behavior of the solution, and the maximum positive eigenvalue and its eigenvector with nonzero entries play a key role in approximating asymptotically the structural balance. Finally, the numerical simulations show the validity of methods in this paper.

A Sharp Sufficient Condition for Sparsity Pattern Recovery

Sufficient number of linear and noisy measurements for exact and approximate sparsity pattern/support set recovery in the high dimensional setting is derived. Although this problem as been addressed in the recent literature, there is still considerable gaps between those results and the exact limits of the perfect support set recovery. To reduce this gap, in this paper, the sufficient condition is enhanced. A specific form of a Joint Typicality decoder is used for the support recovery task. Two performance metrics are considered for the recovery validation; one, which considers exact support recovery, and the other which seeks partial support recovery. First, an upper bound is obtained on the error probability of the sparsity pattern recovery. Next, using the mentioned upper bound, sufficient number of measurements for reliable support recovery is derived. It is shown that the sufficient condition for reliable support recovery depends on three key parameters of the problem; the noise variance, the minimum nonzero entry of the unknown sparse vector and the sparsity level. Simulations are performed for different sparsity rate, different noise variances, and different distortion levels. The results show that for all the mentioned cases the proposed methodology increases convergence rate of upper bound of the error probability of support recovery significantly which leads to a lower error probability bound compared with previously proposed bounds.

On maps preserving rank-one nilpotents

In this paper, we show that a linear, bijective map such that whenever XY = M, for M, N rank-one nilpotents, is of the form for an invertible and a non-zero . We extend this result to show in general that, given any M and N in the ring, , of infinite matrices over a field with finitely many non-zero entries, a linear, bijective map , such that whenever XY = M must satisfy for all and .

A note on generalized companion pencils in the monomial basis

In this paper, we introduce a new notion of generalized companion pencils for scalar polynomials over an arbitrary field expressed in the monomial basis. Our definition is quite general and extends the notions of companion pencil in De Teran et al. (Linear Algebra Appl 459:264–333, 2014), generalized companion matrix in Garnett et al. (Linear Algebra Appl 498:360–365, 2016), and Ma–Zhan companion matrices in Ma and Zhan (Linear Algebra Appl 438: 621–625, 2013), as well as the class of quasi-sparse companion pencils introduced in De Teran and Hernando (INdAM Series, Springer, Berlin, pp 157–179, 2019). We analyze some algebraic properties of generalized companion pencils. We determine their Smith canonical form and we prove that they are all nonderogatory. In the last part of the work we will pay attention to the sparsity of these constructions. In particular, by imposing some natural conditions on its entries, we determine the smallest number of nonzero entries of a generalized companion pencil.

Alternating Sign Matrices: Extensions, König-Properties, and Primary Sum-Sequences

This paper is concerned with properties of permutation matrices and alternating sign matrices (ASMs). An ASM is a square $$(0,\pm 1)$$-matrix such that, ignoring 0’s, the 1’s and $$-1$$’s in each row and column alternate, beginning and ending with a 1. We study extensions of permutation matrices into ASMs by changing some zeros to $$+1$$ or $$-1$$. Furthermore, several properties concerning the term rank and line covering of ASMs are shown. An ASM A is determined by a sum-matrix $$\varSigma (A)$$ whose entries are the sums of the entries of its leading submatrices (so determined by the entries of A). We show that those sums corresponding to the nonzero entries of a permutation matrix determine all the entries of the sum-matrix and investigate some of the properties of the resulting sequence of numbers. Finally, we investigate the lattice-properties of the set of ASMs (of order n), where the partial order comes from the Bruhat order for permutation matrices.

The minrank of random graphs over arbitrary fields

The minrank of a graph G on the set of vertices [n] over a field $$\mathbb{F}$$ is the minimum possible rank of a matrix $$M \in \mathbb{F}^{{n}\times{n}}$$ with nonzero diagonal entries such that Mi,j = 0 whenever i and j are distinct nonadjacent vertices of G. This notion, over the real field, arises in the study of the Lovász theta function of a graph. We obtain tight bounds for the typical minrank of the binomial random graph G(n, p) over any finite or infinite field, showing that for every field $$\mathbb{F}=\mathbb{F}(n)$$ and every p = p(n) satisfying n-1 ≤ p ≤ 1 - n-0.99, the minrank of G = G(n, p) over $$\mathbb{F}$$ is $$\Theta(\frac{n {\rm{log}}(1/p)}{{\rm{log}} n})$$ with high probability. The result for the real field settles a problem raised by Knuth in 1994. The proof combines a recent argument of Golovnev, Regev and Weinstein, who proved the above result for finite fields of size at most nO(1), with tools from linear algebra, including an estimate of Rónyai, Babai and Ganapathy for the number of zero-patterns of a sequence of polynomials.

Globally maximizing the sum of squares of quadratic forms over the unit sphere

We first lift the problem of maximizing the sum of squares of quadratic forms over the unit sphere to an equivalent nonlinear optimization problem, which provides a new standard quadratic programming relaxation. Then we employ a simplicial branch and bound algorithm to globally solve the lifted problem and show that the time-complexity is linear with respect to the number of all nonzero entries of the input matrices under certain conditions. Numerical results demonstrate the efficiency of the new algorithm.

Strong Structural Controllability of Systems on Colored Graphs

This paper deals with structural controllability of leader-follower networks. The system matrix defining the network dynamics is a pattern matrix in which a priori given entries are equal to zero, while the remaining entries take nonzero values. The network is called strongly structurally controllable if for all choices of real values for the nonzero entries in the pattern matrix, the system is controllable in the classical sense. In this paper we introduce a more general notion of strong structural controllability which deals with the situation that given nonzero entries in the system's pattern matrix are constrained to take identical nonzero values. The constraint of identical nonzero entries can be caused by symmetry considerations or physical constraints on the network. The aim of this paper is to establish graph theoretic conditions for this more general property of strong structural controllability.

A Graphical Characterization of Structurally Controllable Linear Systems With Dependent Parameters

One version of the concept of structural controllability defined for single-input systems by Lin and subsequently generalized to multi-input systems by others, states that a parameterized matrix pair $(A, B)$ whose nonzero entries are distinct parameters, is structurally controllable if values can be assigned to the parameters which cause the resulting matrix pair to be controllable. In this paper the concept of structural controllability is broadened to allow for the possibility that a parameter may appear in more than one location in the pair $(A, B)$. Subject to a certain condition on the parameterization called the "binary assumption", an explicit graph-theoretic characterization of such matrix pairs is derived.

Symbolic Evaluation of hp‐FEM Element Matrices

AbstractIn this paper we consider higher order shape functions for finite elements on a triangle. On the reference element Dubiner‐like ansatz functions based on suitable integrated Jacobi polynomials are chosen. It can be proved that the corresponding mass and stiffness matrices are sparse for all polynomial degree p. Due to the orthogonal relations between Jacobi polynomials the exact values of the entries of mass and stiffness matrix can be determined. Using symbolic computation, we can find simple recurrence relations which allow us to compute the remaining nonzero entries in optimal arithmetic complexity.

Multivariate reciprocal inverse Gaussian distributions from the Sabot–Tarrès–Zeng integral

In Sabot and Tarrès (2015), the authors have explicitly computed the integral STZn=∫exp(−〈x,y〉)(detMx)−1∕2dxwhere Mx is a symmetric matrix of order n with fixed non-positive off-diagonal coefficients and with diagonal (2x1,…,2xn). The domain of integration is the part of Rn for which Mx is positive definite. We calculate more generally for b1≥0,…bn≥0 the integral ∫exp−〈x,y〉−12b⊤Mx−1b(detMx)−1∕2dx,we show that it leads to a natural family of distributions in Rn, called the MRIGn probability laws. This family is stable by marginalization and by conditioning, and it has number of properties which are multivariate versions of familiar properties of univariate reciprocal inverse Gaussian distribution. In general, if the power of detMx under the integral in STZn is distinct from −1∕2 it is not known how to compute the integral. However, introducing the graph G having V={1,…,n} for set of vertices and the set E of {i,j}′ s of non-zero entries of Mx as set of edges, we show also that in the particular case where G is a tree, the integral ∫exp(−〈x,y〉)(detMx)q−1dxwhere q>0, is computable in terms of the MacDonald function Kq.

On the Global Geometry of Sphere-Constrained Sparse Blind Deconvolution.

Blind deconvolution is the problem of recovering a convolutional kernel a0 and an activation signal x0 from their convolution [Formula: see text]. This problem is ill-posed without further constraints or priors. This paper studies the situation where the nonzero entries in the activation signal are sparsely and randomly populated. We normalize the convolution kernel to have unit Frobenius norm and cast the sparse blind deconvolution problem as a nonconvex optimization problem over the sphere. With this spherical constraint, every spurious local minimum turns out to be close to some signed shift truncation of the ground truth, under certain hypotheses. This benign property motivates an effective two stage algorithm that recovers the ground truth from the partial information offered by a suboptimal local minimum. This geometry-inspired algorithm recovers the ground truth for certain microscopy problems, also exhibits promising performance in the more challenging image deblurring problem. Our insights into the global geometry and the two stage algorithm extend to the convolutional dictionary learning problem, where a superposition of multiple convolution signals is observed.