Nonnegative Vector Research Articles

A Scalable and Statistically Robust Beam Alignment Technique for Millimeter-Wave Systems

Millimeter-wave (mm-wave) frequency bands provide an opportunity for much wider channel bandwidth compared with the traditional sub-6-GHz band. Communication at mm-waves is, however, quite challenging due to the severe propagation pathloss incurred by conventional isotropic antennas. To cope with this problem, directional beamforming both at the base station (BS) side and at the user equipment (UE) side is necessary in order to establish a strong path conveying enough signal power. Finding such beamforming directions is referred to as beam alignment (BA). This paper presents a new scheme for efficient BA. Our scheme finds a strong propagation path identified by an angle-of-arrival (AoA) and angle-of-departure (AoD) pair, by exploring the AoA–AoD domain through pseudo-random multi-finger beam patterns and constructing an estimate of the resulting second-order statistics (namely, the average received power for each pseudo-random beam configuration). The resulting under-determined system of equations is efficiently solved using non-negative constrained least-squares, yielding naturally a sparse non-negative vector solution whose maximum component identifies the optimal path. As a result, our scheme is highly robust to variations of the channel time dynamics compared with alternative concurrent approaches based on the estimation of the instantaneous channel coefficients, rather than of their second-order statistics. In the proposed scheme, the BS probes the channel in the downlink and trains simultaneously an arbitrarily large number of UEs. Thus, “beam refinement,” with multiple interactive rounds of downlink/uplink transmissions, is not needed. This results in a scalable BA protocol, where the protocol overhead is virtually independent of the number of UEs, since all the UEs run the BA procedure at the same time. Extensive simulation results illustrate that our approach is superior to the state-of-the-art BA schemes proposed in the literature in terms of training overhead in multi-user scenarios and robustness to variations in the channel dynamics.

Ehrhart Series of Fractional Stable Set Polytopes of Finite Graphs

The fractional stable set polytope ${\rm FRAC}(G)$ of a simple graph $G$ with $d$ vertices is a rational polytope that is the set of nonnegative vectors $(x_1,\ldots,x_d)$ satisfying $x_i+x_j\le 1$ for every edge $(i,j)$ of $G$. In this paper we show that (i) The $\delta$-vector of a lattice polytope $2 {\rm FRAC}(G)$ is alternatingly increasing; (ii) The Ehrhart ring of ${\rm FRAC}(G)$ is Gorenstein; (iii) The coefficients of the numerator of the Ehrhart series of ${\rm FRAC}(G)$ are symmetric, unimodal and computed by the $\delta$-vector of $2 {\rm FRAC}(G)$.

Solution of Optimal Harvesting Problem by Finite Difference Approximations of Size-Structured Population Model

We solve numerically a forest management optimization problem governed by a nonlinear partial differential equation (PDE), which is a size-structured population model. The formulated problem is supplemented with a natural constraint for a solution to be non-negative. PDE is approximated by an explicit or implicit in time finite difference scheme, whereas the cost function is taken from the very beginning in the finite-dimensional form used in practice. We prove the stability of the constructed nonlinear finite difference schemes on the set of non-negative vectors and the solvability of the formulated discrete optimal control problems. The gradient information is derived by constructing discrete adjoint state equations. The projected gradient method is used for finding the extremal points. The results of numerical testing for several real problems show good agreement with the known results and confirm the theoretical statements.

A boundary Schwarz lemma for pluriharmonic mappings from the unit polydisk to the unit ball

In this article, we present a Schwarz lemma at the boundary for pluriharmonic mappings from the unit polydisk to the unit ball, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimensions. It is proved that if the pluriharmonic mapping f∈P(Dn,BN) is C1+α at z0∈Er⊂∂Dn with f(0)= 0 and f(z0)=w0∈ ∂BN for any n, N ≥ 1, then there exist a nonnegative vector λf=(λ1, 0, … , λr, 0, …, 0)T∈R2n satisfying λi≥122n-1 for 1≤i≤r such that (Df(z′0))Tw′0=diag(λf)z′0,where z′0 and w′0 are real versions of z0 and w0, respectively.

A linear programming approach for stabilization of positive Markovian jump systems with a saturated single input

This paper proposes a linear programming approach for stabilization of positive Markovian jump systems (PMJSs) with a saturated single input. The proposed approach first derives a sufficient condition for stabilization of PMJSs with input saturation based on the linear co-positive Lyapunov function. By introducing an intermediate scalar whose absolute value is less than the absolute value of product of nonnegative vector of the linear co-positive Lyapunov function and input matrix and constructing a special form of the controller gains, this approach obtains a modified condition applicable for the linear programming. Finally, four numerical examples show that the proposed approach gives the larger domain of attraction than the existing approach based on the quadratic Lyapunov function.

On a Class of Semi-Positive Tensors in Tensor Complementarity Problem

Recently, the tensor complementarity problem has been investigated in the literature. In this paper, we extend a class of structured matrices to higher-order tensors; the corresponding tensor complementarity problem has a unique solution for any nonzero nonnegative vector. We discuss their relationships with semi-positive tensors and strictly semi-positive tensors. We also study the property of such a structured tensor. We show that every principal sub-tensor of such a structured tensor is still a structured tensor in the same class, with a lower dimension. We also give two equivalent formulations of such a structured tensor.

Adaptive multiple graph regularized semi-supervised extreme learning machine

Semi-supervised extreme learning machine (SSELM) was proposed as an effective algorithm for machine learning and pattern recognition. However, the performance of SSELM heavily depends on whether the underlying geometrical structure of the data can be well exploited. Though many techniques have been utilized for constructing graph to represent the data structure, which of them can best reflect the intrinsic distribution of complicated input data is still needed to be verified. Aiming to solve this problem, we propose a novel algorithm called adaptive multiple graph regularized semi-supervised extreme learning machine (AMGR-SSELM). The contributions of the proposed algorithm are as follows: (1) AMGR-SSELM employs multiple graph structures extracted from training samples to characterize the structure of input data. Since these graphs are constructed based on different principles and complementary with each other, the underlying data distribution can be well exploited through combining them. (2) A nonnegative weight vector is introduced into AMGR-SSELM to adaptively combine the multiple graphs for representing different data. (3) An explicit classifier can be learnt in our algorithm, which overcomes the ‘out of sample’ problem. (4) A simple and efficient iterative update approach is also proposed to optimize AMGR-SSELM. In addition, we compare the proposed approach with other classification methods and some extreme learning machine variants on five benchmark image databases (Yale, Extended YaleB, CMU PIE, AR and FKP). The results of extensive experiments show the advantages and effectiveness of the proposed approach.

A note on linear preservers of semipositive and minimally semipositive matrices

Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory. In this short note, the structure of linear maps which preserve the set of all semipositive/minimally semipositive matrices is studied. An open problem is solved, and some ambiguities in the article [J. Dorsey, T. Gannon, N. Jacobson, C.R. Johnson and M. Turnansky. Linear preservers of semi-positive matrices. {\em Linear and Multilinear Algebra}, 64:1853--1862, 2016.] are clarified.

New Contributions to Semipositive and Minimally Semipositive Matrices

Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory. In this article, this notion is revisited and new results are presented. It is shown that the set of all $m \times n$ minimally semipositive matrices contains a basis for the linear space of all $m \times n$ matrices. Apart from considerations involving principal pivot transforms and the Schur complement, results on semipositivity and/or minimal semipositivity for the following classes of matrices are presented: intervals of rectangular matrices, skew-symmetric and almost skew-symmetric matrices, copositive matrices, $N$-matrices, almost $N$-matrices and almost $P$-matrices.

Extremal Copositive Matrices with Zero Supports of Cardinality n-2

Let $A \in {\cal C}^n$ be an exceptional extremal copositive $n \times n$ matrix with positive diagonal. A zero $u$ of $A$ is a non-zero nonnegative vector such that $u^TAu = 0$. The support of a zero $u$ is the index set of the positive elements of $u$. A zero $u$ is minimal if there is no other zero $v$ such that $\Supp v \subset \Supp u$ strictly. Let $G$ be the graph on $n$ vertices which has an edge $(i,j)$ if and only if $A$ has a zero with support $\{1,\dots,n\} \setminus \{i,j\}$. In this paper, it is shown that $G$ cannot contain a cycle of length strictly smaller than $n$. As a consequence, if all minimal zeros of $A$ have support of cardinality $n - 2$, then $G$ must be the cycle graph $C_n$.

Extremal Copositive Matrices with Zero Supports of Cardinality n-2

Let $A \in {\cal C}^n$ be an exceptional extremal copositive $n \times n$ matrix with positive diagonal. A zero $u$ of $A$ is a non-zero nonnegative vector such that $u^TAu = 0$. The support of a zero $u$ is the index set of the positive elements of $u$. A zero $u$ is minimal if there is no other zero $v$ such that $\Supp v \subset \Supp u$ strictly. Let $G$ be the graph on $n$ vertices which has an edge $(i,j)$ if and only if $A$ has a zero with support $\{1,\dots,n\} \setminus \{i,j\}$. In this paper, it is shown that $G$ cannot contain a cycle of length strictly smaller than $n$. As a consequence, if all minimal zeros of $A$ have support of cardinality $n - 2$, then $G$ must be the cycle graph $C_n$.

Robust Nonnegative Sparse Recovery and the Nullspace Property of 0/1 Measurements

We investigate recovery of nonnegative vectors from non-adaptive compressive measurements in the presence of noise of unknown power. In the absence of noise, existing results in the literature identify properties of the measurement that assure uniqueness in the non-negative orthant. By linking such uniqueness results to nullspace properties, we deduce uniform and robust compressed sensing guarantees for nonnegative least squares. No $\ell _{1}$ -regularization is required. As an important proof of principle, we establish that $m\times n$ random i.i.d. 0/1-valued Bernoulli matrices obey the required conditions with overwhelming probability provided that $m= \mathcal {O}(s\log (n/s))$ . We achieve this by establishing the robust nullspace property for random 0/1-matrices—a novel result in its own right. Our analysis is motivated by applications in wireless network activity detection.

Separability and decomposition in mechanism design with transfers

In private values quasi-linear environment, we consider problems where allocation decisions along multiple components have to be made. Every agent has additively separable valuation over the components. We show that every unanimous and dominant strategy implementable allocation rule in this problem is a component-wise weighted utilitarian rule, which assigns non-negative weight vectors to agents in each component and chooses an alternative in each component by maximizing the weighted sum of valuations in that component. A corollary of our result is that every unanimous and dominant strategy implementable allocation rule can be almost decomposed (modulo tie-breaking) into dominant strategy implementable allocation rules along each component.

A multi-view-group non-negative matrix factorization approach for automatic image annotation

In automatic image annotation (AIA) different features describe images from different aspects or views. Part of information embedded in some views is common for all views, while other parts are individual and specific. In this paper, we present the Mvg-NMF approach, a multi-view-group non-negative matrix factorization (NMF) method for an AIA system which considers both common and individual factors. The NMF framework discovers a latent space by decomposing data into a set of non-negative basis vectors and coefficients. The views divided into homogeneous groups and latent spaces are extracted for each group. After mapping the test images into these spaces, a unified distance matrix is computed from the distance between images in all spaces. Then a search-based method is used to propagate tags from the nearest neighbors to test images. The evaluation on three datasets commonly used for image annotation showed that the Mvg-NMF is highly competitive with the recent state-of-the-art works.

A generalized approximation framework for fractional network flow and packing problems

We generalize the fractional packing framework of Garg and Koenemann (SIAM J Comput 37(2):630–652, 2007) to the case of linear fractional packing problems over polyhedral cones. More precisely, we provide approximation algorithms for problems of the form $$\max \{c^T x : Ax \le b, x \in C \}$$ , where the matrix A contains no negative entries and C is a cone that is generated by a finite set S of non-negative vectors. While the cone is allowed to require an exponential-sized representation, we assume that we can access it via one of three types of oracles. For each of these oracles, we present positive results for the approximability of the packing problem. In contrast to other frameworks, the presented one allows the use of arbitrary linear objective functions and can be applied to a large class of packing problems without much effort. In particular, our framework instantly allows to derive fast and simple fully polynomial-time approximation algorithms (FPTASs) for a large set of network flow problems, such as budget-constrained versions of traditional network flows, multicommodity flows, or generalized flows. Some of these FPTASs represent the first ones of their kind, while others match existing results but offer a much simpler proof.

Evolutionary nonnegative matrix factorization with adaptive control of cluster quality

Nonnegative matrix factorization (NMF) approximates a given data matrix using linear combinations of a small number of nonnegative basis vectors, weighted by nonnegative encoding coefficients. This enables the exploration of the cluster structure of the data through the examination of the values of the encoding coefficients and therefore, NMF is often used as a popular tool for clustering analysis. However, its encoding coefficients do not always reveal a satisfactory cluster structure. To improve its effectiveness, a novel evolutionary strategy is proposed here to drive the iterative updating scheme of NMF and generate encoding coefficients of higher quality that are capable of offering more accurate and sharper cluster structures. The proposed hybridization procedure that relies on multiple initializations reinforces the robustness of the solution. Additionally, three evolving rules are designed to simultaneously boost the cluster quality and the reconstruction error during the iterative updates. Any clustering performance measure, such as either an internal one relying on the data itself or an external based on the availability of ground truth information, can be employed to drive the evolving procedure. The effectiveness of the proposed method is demonstrated via careful experimental designs and thorough comparative analyses using multiple benchmark datasets.

The Solvency II Standard Formula, Linear Geometry, and Diversification

The core of risk aggregation in the Solvency II Standard Formula is the so-called square root formula. We argue that it should be seen as a means for the aggregation of different risks to an overall risk rather than being associated with variance-covariance based risk analysis. Considering the Solvency II Standard Formula from the viewpoint of linear geometry, we immediately find that it defines a norm and therefore provides a homogeneous and sub-additive tool for risk aggregation. Hence, Euler’s Principle for the reallocation of risk capital applies and yields explicit formulas for capital allocation in the framework given by the Solvency II Standard Formula. This gives rise to the definition of diversification functions, which we define as monotone, subadditive, and homogeneous functions on a convex cone. Diversification functions constitute a class of models for the study of the aggregation of risk and diversification. The aggregation of risk measures using a diversification function preserves the respective properties of these risk measures. Examples of diversification functions are given by seminorms, which are monotone on the convex cone of non-negative vectors. Each L p norm has this property, and any scalar product given by a non-negative positive semidefinite matrix does as well. In particular, the Standard Formula is a diversification function and hence a risk measure that preserves homogeneity, subadditivity and convexity.

Underdetermined DOA Estimation Method for Wideband Signals Using Joint Nonnegative Sparse Bayesian Learning

Underdetermined direction-of-arrival (DOA) estimation for wideband signals by sparse arrays is discussed in the framework of sparse Bayesian learning (SBL). The problem is transformed to recovering multiple nonnegative sparse vectors, which share the same sparse support but correspond to distinct overcomplete basis matrices, from their noise contaminated linear combination vectors. A two-layer Bayesian model is established, and a hyperparameter vector, which reveals the true DOAs, is set to control this common sparsity in the model. The expectation-maximization algorithm is employed to realize this joint nonnegative SBL procedure, which can give the DOA estimation in a few iterations. The proposed method manifests mild computational complexity, and numerical simulation results show that compared with the existing underdetermined DOA estimation methods, it yields superior estimation accuracy, without the prior knowledge of number of sources.

A sparse fast Fourier algorithm for real non-negative vectors

In this paper we propose a new fast Fourier transform to recover a real non-negative signal x∈R+N from its discrete Fourier transform x̂=FNx∈CN. If the signal x appears to have a short support, i.e., vanishes outside a support interval of length m<N, then the algorithm has an arithmetical complexity of only O(mlogmlog(N/m)) and requires O(mlog(N/m)) Fourier samples for this computation. In contrast to other approaches there is no a priori knowledge needed about sparsity or support bounds for the vector x. The algorithm automatically recognizes and exploits a possible short support of the vector and falls back to a usual radix-2 FFT algorithm if x has (almost) full support. The numerical stability of the proposed algorithm is shown by numerical examples.

Precise large deviations for sums of random vectors with dependent components of consistently varying tails

Let {X i = (X 1,i ,...,X m,i )⊤, i ≥ 1} be a sequence of independent and identically distributed nonnegative m-dimensional random vectors. The univariate marginal distributions of these vectors have consistently varying tails and finite means. Here, the components of X 1 are allowed to be generally dependent. Moreover, let N(·) be a nonnegative integer-valued process, independent of the sequence {X i , i ≥ 1}. Under several mild assumptions, precise large deviations for S n = Σ i=1 n X i and S N(t) = Σ i=1 N(t) X i are investigated. Meanwhile, some simulation examples are also given to illustrate the results.