Nonnegative Entries Research Articles

Chinese Character Recognition Using Non-negative Matrix Factorization

Non-negative matrix factorization (NMF) was introduced by Paatero and Tapper in 1994 and it was a general way of reducing the dimension of the matrix with non-negative entries. Non-negative matrix factorization is very useful in many data analysis applications such as character recognition, text mining, and others. This paper aims to study the application in Chinese character recognition using non-negative matrix factorization. Python was used to carry out the LU factorization and non-negative matrix factorization of a Chinese character in Boolean Matrix. Preliminary analysis confirmed that the data size of and and are chosen for the NMF of the Boolean matrix. In this project, one hundred printed Chinese characters were selected, and all the Chinese characters can be categorized into ten categories according to the number of strokes , for . The Euclidean distance between the Boolean matrix of a Chinese character and the matrix after both LU factorization and NMF is calculated for further analysis. Paired t-test confirmed that the factorization of Chinese characters in the Boolean matrix using NMF is better than the LU factorization. Finally, ten handwritten Chinese characters were selected to test whether the program is able to identify the handwritten and the printed Chinese characters. Experimental results showed that 70% of the characters can be recognized via the least Euclidean distance obtained. NMF is suitable to be applied in Chinese character recognition since it can reduce the dimension of the image and the error between the original Boolean matrix and after NMF is less than 5%.

Approximation Scheme for Weighted Metric Clustering via Sherali-Adams

Motivated by applications to classification problems on metric data, we study Weighted Metric Clustering problem: given a metric d over n points and a k x k symmetric matrix A with non-negative entries, the goal is to find a k-partition of these points into clusters C1,...,Ck, while minimizing the sum of A[i,j] * d(u,v) over all pairs of clusters Ci and Cj and all pairs of points u from Ci and v from Cj. Specific choices of A lead to Weighted Metric Clustering capturing well-studied graph partitioning problems in metric spaces, such as Min-Uncut, Min-k-Sum, Min-k-Cut, and more. Our main result is that Weighted Metric Clustering admits a polynomial-time approximation scheme (PTAS). Our algorithm handles all the above problems using the Sherali-Adams linear programming relaxation. This subsumes several prior works, unifies many of the techniques for various metric clustering objectives, and yields a PTAS for several new problems, including metric clustering on manifolds and a new family of hierarchical clustering objectives. Our experiments on the hierarchical clustering objective show that it better captures the ground-truth structural information compared to the popular Dasgupta's objective.

A Riordan group poset

In this paper, we look at Riordan arrays with nonnegative integer entries. In the set, if AT=B for a nonnegative integer matrix T, then we say A≤B. Our primary focus centers around six kinds of key arrays: Appell, Bell, two-Bell, Natural, Hankel, and Pseudo-involution. We examine relations among Hankel, Bell and two-Bell in general, while the other entries are treated case by case. We start by looking at individual cases related to the little and large Schröder numbers. Then we look at the families related to k-Catalan arrays, k-Schröder arrays, and k-Motzkin arrays. We often can give combinatorial interpretations. Along the way, several intriguing counterexamples occur.

A Liouville type theorem for a scaling invariant parabolic system with no gradient structure

We prove a Liouville-type theorem for semilinear parabolic systems of the form∂tui−Δui=∑j=1mβijuirujp−r,i=1,2,...,m in the whole space RN×R, where r>0, p>r+1 and (βij) is a real (m,m) symmetric matrix with nonnegative entries, positive on the diagonal. Due to the lack of gradient structure, a nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron is introduced. We expect to see more applications of this method to other problems with no gradient structure.

A Sequence on the Maximum Eigenvalue of a Nonnegative Matrix

For an irreducible matrix with nonnegative entries, a sequence is constructed on the matrix trace. And we prove the convergence of the sequence in two cases when the irreducible matrix with nonnegative entries is a primitive matrix and a non-primitive matrix.

Pattern-Multiplicative Average of Nonnegative Matrices Revisited: Eigenvalue Approximation Is the Best of Versatile Optimization Tools

Given several nonnegative matrices with a single pattern of allocation among their zero/nonzero elements, the average matrix should have the same pattern, too. This is the first tenet of the pattern-multiplicative average (PMA) concept, while the second one suggests the multiplicative (or geometric) nature of averaging. The original concept of PMA was motivated by the practice of matrix population models as a tool to assess the population viability from long-term monitoring data. The task has reduced to searching for an approximate solution to an overdetermined system of polynomial equations for unknown elements of the average matrix (G), and hence to a nonlinear constrained minimization problem for the matrix norm. Former practical solutions faced certain technical problems, which required sophisticated algorithms but returned acceptable estimates. Now, we formulate (for the first time in ecological modeling and nonnegative matrix theory) the PMA problem as an eigenvalue approximation one and reduce it to a standard problem of linear programing (LP). The basic equation of averaging also determines the exact value of λ1(G), the dominant eigenvalue of matrix G, and the corresponding eigenvector. These are bound by the well-known linear equations, which enable an LP formulation of the former nonlinear problem. The LP approach is realized for 13 fixed-pattern matrices gained in a case study of Androsace albana, an alpine short-lived perennial, monitored on permanent plots over 14 years. A standard software routine reveals the unique exact solution, rather than an approximate one, to the PMA problem, which turns the LP approach into ‘’the best of versatile optimization tools”. The exact solution turns out to be peculiar in reaching zero bounds for certain nonnegative entries of G, which deserves modified problem formulation separating the lower bounds from zero.

Automatic selfadjoint-ideal semigroups for finite matrices

The notion of automatic selfadjointness of all ideals in a multiplicative semigroup of the bounded linear operators on a separable Hilbert space B(H) arose in a 2015 discussion with Heydar Radjavi who pointed out that B(H) and the finite rank operators F(H) possessed this unitary invariant property which category we named SI semigroups (for automatic selfadjoint ideal semigroups). Equivalent to the SI property is the solvability, for each A in the semigroup, of the bilinear operator equation A⁎=XAY which we believe is a new connection relating the semigroup theory with the theory of operator equations.We found in our earlier works in the subject that even at the basic level of singly generated semigroups, the investigation of SI semigroups led to interesting algebraic and analytic phenomena when generated by rank one operators, normal operators, partial and power partial isometries, subnormal-hyponormal-essentially normal operators, and weighted shift operators; and generated by commuting families of normal operators.In this paper, we focus on a separate Mn(C) treatment for singly generated SI semigroups that requires studying the solvability of the bilinear matrix equation A⁎=XAY in a multiplicative semigroup of finite matrices. This separate focus is needed because the techniques employed in our earlier works we could not adapt to finite matrices. In this paper we find that for certain classes of generators, being a partial isometry is equivalent to generating an SI semigroup. Such classes are: degree 2 nilpotent matrices, weighted shifts, and non-normal Jordan matrices. For the key tools used to establish these equivalences, we developed a number of necessary conditions for singly generated semigroups to be SI for the very general classes: nonselfadjoint matrices, nonzero nilpotent matrices, nonselfadjoint invertible matrices, and Jordan blocks. We also show, for a nonselfadjoint matrix generator in an SI semigroup, the matrix being a partial isometry is equivalent to having norm one. And as an aside, we also prove necessary generator conditions for the SI property when generated by matrices with nonnegative entries.

Random projections of linear and semidefinite problems with linear inequalities

The Johnson-Lindenstrauss Lemma states that there exist linear maps that project a set of points of a vector space into a space of much lower dimension such that the Euclidean distance between these points is approximately preserved. This lemma has been previously used to prove that we can randomly aggregate, using a random matrix whose entries are drawn from a zero-mean sub-Gaussian distribution, the equality constraints of an Linear Program (LP) while preserving approximately the value of the problem. In this paper we extend these results to the inequality case by introducing a random matrix with non-negative entries that allows to randomly aggregate inequality constraints of an LP while preserving approximately the value of the problem. By duality, the approach we propose allows to reduce both the number of constraints and the dimension of the problem while obtaining some theoretical guarantees on the optimal value. We will also show an extension of our results to certain semidefinite programming instances.

On Hadamard powers of positive semi-definite matrices

Consider the set of scalars α \alpha for which the α \alpha th Hadamard power of any n × n n\times n positive semi-definite (p.s.d.) matrix with non-negative entries is p.s.d. It is known that this set is of the form { 0 , 1 , … , n − 3 } ∪ [ n − 2 , ∞ ) \{0, 1, \dots , n-3\}\cup [n-2, \infty ) . A natural question is “what is the possible form of the set of such α \alpha for a fixed p.s.d. matrix with non-negative entries?”. In all examples appearing in the literature, the set turns out to be union of a finite set and a semi-infinite interval. In this article, examples of matrices are given for which the set consists of a finite set and more than one disjoint interval of positive length. In fact, it is proved that the number of such disjoint intervals can be made arbitrarily large, by giving explicit examples of matrices. The case when the entries of the matrices are not necessarily non-negative is also considered.

Ehrhart Positivity of Tesler Polytopes and Berline–Vergne’s Valuation

For {varvec{a}} in {mathbb {R}}_{ge 0}^{n}, the Tesler polytope {text {Tes}}_{n}({varvec{a}}) is the set of upper triangular matrices with non-negative entries whose hook sum vector is {varvec{a}}. Motivated by a conjecture of Morales, we study the questions of whether the coefficients of the Ehrhart polynomial of {text {Tes}}_n(1,1,dots ,1) are positive. We attack this problem by studying a certain function constructed by Berline–Vergne and its values on faces of a unimodularly equivalent copy of {text {Tes}}_n(1,1,dots ,1). We develop a method of obtaining the dot products appeared in formulas for computing Berline–Vergne’s function directly from facet normal vectors. Using this method together with known formulas, we are able to show Berline–Vergne’s function has positive values on codimension 2 and 3 faces of the polytopes we consider. As a consequence, we prove that the third and fourth coefficients of the Ehrhart polynomial of {text {Tes}}_{n}(1,dots ,1) are positive. Using the Reduction Theorem by Castillo and the second author, we generalize the above result to all deformations of {text {Tes}}_{n}(1,dots ,1) including all the integral Tesler polytopes.

The first eigenvector of a distance matrix is nearly constant

Let x1,…,xn be points in a metric space and define the distance matrix D∈Rn×n by Dij=d(xi,xj). The Perron-Frobenius Theorem implies that there is an eigenvector v∈Rn with non-negative entries associated to the largest eigenvalue. We prove that this eigenvector is nearly constant in the sense that the inner product with the constant vector 1∈Rn is large〈v,1〉≥12⋅‖v‖ℓ2⋅‖1‖ℓ2 and that each entry satisfies vi≥‖v‖ℓ2/4n. Both inequalities are sharp.

Linear mappings preserving the completely positive rank

Let Sn be the set of all n-by-n symmetric real matrices. A matrix a∈Sn has completely positive rank (cp rank) equal to k if a is the sum of k symmetric rank-one matrices with all entries nonnegative, and k is the smallest number that allows such a representation. A mapping φ:Sn→Sn is a cp rank preserver if, for every k and every n×n matrix a of cp rank k, the image φ(a) has cp rank equal to k. We prove that, for all n⩾6, every linear cp rank preserver φ:Sn→Sn has the form φ(x)=m⊤xm for a fixed monomial matrix m with nonnegative entries.

Limiting eigenvalue behavior of a class of large dimensional random matrices formed from a Hadamard product

This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices [Formula: see text], studied in [V. L. Girko, Theory of Stochastic Canonical Equations: Vol. [Formula: see text] (Kluwer Academic Publishers, Dordrecht, 2001)]. Here, [Formula: see text] is an [Formula: see text] random matrix consisting of independent complex standardized random variables, [Formula: see text], [Formula: see text], has nonnegative entries, and ∘ denotes Hadamard (componentwise) product. Results are obtained under assumptions on the entries of [Formula: see text] and [Formula: see text] which are different from those in [V. L. Girko, Theory of Stochastic Canonical Equations: Vol. 1 (Kluwer Academic Publishers, Dordrecht, 2001)], which include a Lindeberg condition on the entries of [Formula: see text], as well as a bound on the average of the rows and columns of [Formula: see text]. The present paper separates the assumptions needed on [Formula: see text] and [Formula: see text]. It assumes a Lindeberg condition on the entries of [Formula: see text], along with a tightness-like condition on the entries of [Formula: see text].

A new perspective on cooperative control of multi-agent systems through different types of graph Laplacians

In the field of systems and control, many cooperative control problems of multi-agent systems have been actively studied in the past two decades. This article aims to organize extensive existing work on different cooperative control problems into three categories, based on three different types of graph Laplacian matrices involved. A Laplacian matrix is an important representation of graph topology, and depending on the field of its entries, there are three types: ordinary Laplacian (non-negative diagonal entries and non-positive off-diagonal entries), signed Laplacian (arbitrary real entries), and complex Laplacian (arbitrary complex entries). Each type of graph Laplacian is useful in modeling and solving a different set of cooperative control problems. In particular, their algebraic properties are fundamental in characterizing stability and performance of the respective solution algorithms. To our best knowledge, organizing the literature on multi-agent cooperative control through the lens of different graph Laplacians is new.

Determinants and limit systems in some idempotent and non-associative algebraic structure

This paper considers an idempotent and symmetrical algebraic structure as well as some closely related concepts. A special notion of determinant is introduced and a Cramer formula is derived for a class of limit systems derived from the Hadamard matrix product. Thereby, some standard results arising for Max-Times systems with nonnegative entries appear as a special case. The case of two sided systems is also analyzed. In addition, a notion of eigenvalue in limit is considered. It is shown that one can construct a special semi-continuous regularized polynomial whose zeros are related to the eigenvalues of a matrix with nonnegative entries.

Deep alternating non-negative matrix factorisation

Non-negative matrix factorisation (NMF) is a promising data-mining technique for non-negative data. NMF achieves feature extraction by factorising the original data matrix into a basis matrix and coding matrix both with non-negative entries. Recently, multi-layer or deep NMF has been studied because of its ability to extract deep representative features which can help profoundly understand the original data. The existing deep NMF approaches are implemented by cascading the factorisation of the coding matrix at each layer. This paper proposes a novel scheme, in which the factorisations of the basis and coding matrices are alternated along layers. Based on this scheme, several deep NMF models to address various data scenarios are developed. Extensive experimental results on several machine learning tasks, including collaborative filtering, image inpainting, and community detection, show that the proposed alternating deep factorisation algorithms perform significantly better than existing state-of-the-art algorithms in most instances of these tasks.

A Fractal Eigenvector

The recursively-constructed family of Mandelbrot matrices for n = 1, 2, have nonnegative entries (indeed just 0 and 1, so each can be called a binary matrix) and have eigenvalues whose negatives give periodic orbits under the Mandelbrot iteration, namely with , and are thus contained in the Mandelbrot set. By the Perron–Frobenius theorem, the matrices have a dominant real positive eigenvalue, which we call ρn . This article examines the eigenvector belonging to that dominant eigenvalue and its fractal-like structure, and similarly examines (with less success) the dominant singular vectors of from the singular value decomposition.

Further enumeration results concerning a recent equivalence of restricted inversion sequences

Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(\geq,\neq,>)$ is the same as that of $n-1-\text{asc}$ on the class $I_n(>,\neq,\geq)$, which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on $I_n(\geq,\neq,>)$ of asc and desc along with two other parameters, and do the same for $n-1-\text{asc}$ and desc on $I_n(>,\neq,\geq)$. By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and provides a new proof of) the recent result $|I_n(\geq,\neq,>)|=|I_n(>,\neq,\geq)|$. In both cases, an algorithm is formulated for computing the generating function of the asc distribution on members of each respective class having a fixed number of descents.

Positivity of Hadamard powers of a few band matrices

Let $\mathbb{P}_G([0,\infty))$ and $\mathbb{P}_G^{'}([0,\infty))$ be the sets of positive semidefinite and positive definite matrices of order $n$, respectively, with nonnegative entries, where some positions of zero entries are restricted by a simple graph $G$ with $n$ vertices. It is proved that for a connected simple graph $G$ of order $n\geq 3$, the set of powers preserving positive semidefiniteness on $\mathbb{P}_G([0,\infty))$ is precisely the same as the set of powers preserving positive definiteness on $\mathbb{P}_G^{'}([0,\infty))$. In particular, this provides an explicit combinatorial description of the critical exponent for positive definiteness, for all chordal graphs. Using chain sequences, it is proved that the Hadamard powers preserving the positive (semi) definiteness of every tridiagonal matrix with nonnegative entries are precisely $r\geq 1$. The infinite divisibility of tridiagonal matrices is studied. The same results are proved for a special family of pentadiagonal matrices.

Rees algebras of filtrations of covering polyhedra and integral closure of powers of monomial ideals

The aims of this work are to study Rees algebras of filtrations of monomial ideals associated with covering polyhedra of rational matrices with nonnegative entries and nonzero columns using combinatorial optimization and integer programming and to study powers of monomial ideals and their integral closures using irreducible decompositions and polyhedral geometry. We study the Waldschmidt constant and the ic-resurgence of the filtration associated with a covering polyhedron and show how to compute these constants using linear programming. Then, we show a lower bound for the ic-resurgence of the ideal of covers of a graph and prove that the lower bound is attained when the graph is perfect. We also show lower bounds for the ic-resurgence of the edge ideal of a graph and give an algorithm to compute the asymptotic resurgence of squarefree monomial ideals. A classification of when Newton’s polyhedron is the irreducible polyhedron is presented using integral closure.