Exact results for perturbation to total positivity and to total nonsingularity

2014 ◽  
Vol 27 ◽  
Author(s):  
Miriam Farber ◽  
Mitchell Faulk ◽  
Charles Johnson ◽  
Evan Marzion
Keyword(s):  
Nonnegative Matrix ◽  
Total Positivity ◽  
Projective Planes ◽  
Singular Matrix ◽  
Line Geometry ◽  
Positive Matrix ◽  
Totally Positive ◽  
Point Line ◽  
Totally Nonnegative Matrix ◽  
The Relationship

A study of the maximum number of equal entries in totally positive and totally nonsingular m-by-n, matrices for small values of m and n, is presented. Equal entries correspond to entries of the totally nonnegative matrix J that are not changed in producing a TP or TNS matrix. It is shown that the maximum number of equal entries in a 7-by-7 totally positive matrix is strictly smaller than that for a 7-by-7 totally non-singular matrix, but, this is the first pair (m; n) for which these maximum numbers differ. Using point-line geometry in the projective plane, a family of values for (m; n) for which these maximum numbers differ is presented. Generalization to the Hadamard core, as well as larger projective planes is also established. Finally, the relationship with C4 free graphs, along with a method for producing symmetric TP matrices with maximal symmetric arrangements of equal entries is discussed.

Factor-Bounded Nonnegative Matrix Factorization

2021 ◽  
Vol 15 (6) ◽  
pp. 1-18
Author(s):  
Kai Liu ◽  
Xiangyu Li ◽  
Zhihui Zhu ◽  
Lodewijk Brand ◽  
Hua Wang
Keyword(s):  
Matrix Factorization ◽  
Clustering Algorithm ◽  
Nonnegative Matrix Factorization ◽  
Nonnegative Matrix ◽  
Optimization Methods ◽  
Auxiliary Function ◽  
Image Clustering ◽  
Real World Datasets ◽  
The Relationship ◽  
Matrix Factors

Nonnegative Matrix Factorization (NMF) is broadly used to determine class membership in a variety of clustering applications. From movie recommendations and image clustering to visual feature extractions, NMF has applications to solve a large number of knowledge discovery and data mining problems. Traditional optimization methods, such as the Multiplicative Updating Algorithm (MUA), solves the NMF problem by utilizing an auxiliary function to ensure that the objective monotonically decreases. Although the objective in MUA converges, there exists no proof to show that the learned matrix factors converge as well. Without this rigorous analysis, the clustering performance and stability of the NMF algorithms cannot be guaranteed. To address this knowledge gap, in this article, we study the factor-bounded NMF problem and provide a solution algorithm with proven convergence by rigorous mathematical analysis, which ensures that both the objective and matrix factors converge. In addition, we show the relationship between MUA and our solution followed by an analysis of the convergence of MUA. Experiments on both toy data and real-world datasets validate the correctness of our proposed method and its utility as an effective clustering algorithm.

Characterization and Coordination of Point-line Trajectories

2004 ◽  
Vol 127 (3) ◽  
pp. 502-505 ◽  
Author(s):  
Kwun-Lon Ting ◽  
Yi Zhang ◽  
Ruj Bunduwongse
Keyword(s):  
Orientation Relationship ◽  
Complete Characterization ◽  
Free Form ◽  
Directed Line ◽  
Free Form Surface ◽  
Point Line ◽  
Line Trajectory ◽  
The Relationship ◽  
Line Direction

A point-line refers to a rigid combination of a directed line and an endpoint along the line. To trace a point-line trajectory, one must control not only the trajectory of the endpoint (the directrix) but also the direction of the point-line (the indicatrix). This paper addresses three issues on point-line trajectories. First of all, by considering the relationship between the point trajectory and the corresponding point-line direction, it offers the complete characterization of point-line trajectories. It presents the coordination of the point-line axis with a free point trajectory and also with a point trajectory constrained on a free form surface. The issue of maintaining an invariant orientation relationship between the point-line axis and the point trajectory is also addressed.

scholarly journals Characterizations of the $G_2(4)$ and $L_3(4)$ Near Octagons

10.37236/8476 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Generalized Quadrangle ◽  
Line Geometry ◽  
Near Polygon ◽  
Generalized Polygon ◽  
Generalized Hexagon ◽  
Split Cayley Hexagon ◽  
Point Line ◽  
Line Spread

A triple $(\mathcal{S},S,\mathcal{Q})$ consisting of a near polygon $\mathcal{S}$, a line spread $S$ of $\mathcal{S}$ and a set $\mathcal{Q}$ of quads of $\mathcal{S}$ is called a polygonal triple if certain nice properties are satisfied, among which there is the requirement that the point-line geometry $\mathcal{S}'$ formed by the lines of $S$ and the quads of $\mathcal{Q}$ is itself also a near polygon. This paper addresses the problem of classifying all near polygons $\mathcal{S}$ that admit a polygonal triple $(\mathcal{S},S,\mathcal{Q})$ for which a given generalized polygon $\mathcal{S}'$ is the associated near polygon. We obtain several nonexistence results and show that the $G_2(4)$ and $L_3(4)$ near octagons are the unique near octagons that admit polygonal triples whose quads are isomorphic to the generalized quadrangle $W(2)$ and whose associated near polygons are respectively isomorphic to the dual split Cayley hexagon $H^D(4)$ and the unique generalized hexagon of order $(4,1)$.

scholarly journals Covariance Matrix Estimation under Total Positivity for Portfolio Selection*

2020 ◽  
Author(s):  
Raj Agrawal ◽  
Uma Roy ◽  
Caroline Uhler
Keyword(s):  
Covariance Matrix ◽  
Total Positivity ◽  
General Purpose ◽  
Shrinkage Estimators ◽  
Positive Dependence ◽  
Matrix Estimation ◽  
Sample Covariance ◽  
Totally Positive ◽  
Markowitz Portfolio ◽  
Previous State

Abstract Selecting the optimal Markowitz portfolio depends on estimating the covariance matrix of the returns of N assets from T periods of historical data. Problematically, N is typically of the same order as T, which makes the sample covariance matrix estimator perform poorly, both empirically and theoretically. While various other general-purpose covariance matrix estimators have been introduced in the financial economics and statistics literature for dealing with the high dimensionality of this problem, we here propose an estimator that exploits the fact that assets are typically positively dependent. This is achieved by imposing that the joint distribution of returns be multivariate totally positive of order 2 (MTP2). This constraint on the covariance matrix not only enforces positive dependence among the assets but also regularizes the covariance matrix, leading to desirable statistical properties such as sparsity. Based on stock market data spanning 30 years, we show that estimating the covariance matrix under MTP2 outperforms previous state-of-the-art methods including shrinkage estimators and factor models.

scholarly journals Laguerre geometries and some connections to generalized quadrangles

2007 ◽  
Vol 83 (3) ◽  
pp. 335-356
Author(s):  
Matthew R. Brown
Keyword(s):  
Three Dimensional ◽  
Line Pair ◽  
Generalized Quadrangles ◽  
Constant Number ◽  
Unique Point ◽  
Line Geometry ◽  
Laguerre Geometry ◽  
Point Line ◽  
Dimensional Projective Space ◽  
Unique Circle

AbstractA Laguerre plane is a geometry of points, lines and circles where three pairwise non-collinear points lie on a unique circle, any line and circle meet uniquely and finally, given a circle C and a point Q not on it for each point P on C there is a unique circle on Q and touching C at P. We generalise to a Laguerre geometry where three pairwise non-collinear points lie on a constant number of circles. Examples and conditions on the parameters of a Laguerre geometry are given.A generalized quadrangle (GQ) is a point, line geometry in which for a non-incident point, line pair (P. m) there exists a unique point on m collinear with P. In certain cases we construct a Laguerre geometry from a GQ and conversely. Using Laguerre geometries we show that a GQ of order (s. s2) satisfying Property (G) at a pair of points is equivalent to a configuration of ovoids in three-dimensional projective space.

Optimal couplings are totally positive and more

2004 ◽  
Vol 41 (A) ◽  
pp. 321-332 ◽  
Author(s):  
Paul Glasserman ◽  
David D. Yao
Keyword(s):  
Transportation Problem ◽  
Random Variables ◽  
Total Positivity ◽  
Bivariate Distribution ◽  
Discrete Distributions ◽  
Optimal Solutions ◽  
Totally Positive ◽  
Maximal Correlation ◽  
Optimal Coupling

An optimal coupling is a bivariate distribution with specified marginals achieving maximal correlation. We show that optimal couplings are totally positive and, in fact, satisfy a strictly stronger condition we call the nonintersection property. For discrete distributions we illustrate the equivalence between optimal coupling and a certain transportation problem. Specifically, the optimal solutions of greedily-solvable transportation problems are totally positive, and even nonintersecting, through a rearrangement of matrix entries that results in a Monge sequence. In coupling continuous random variables or random vectors, we exploit a characterization of optimal couplings in terms of subgradients of a closed convex function to establish a generalization of the nonintersection property. We argue that nonintersection is not only stronger than total positivity, it is the more natural concept for the singular distributions that arise in coupling continuous random variables.

Optimal couplings are totally positive and more

2004 ◽  
Vol 41 (A) ◽  
pp. 321-332
Author(s):  
Paul Glasserman ◽  
David D. Yao
Keyword(s):  
Transportation Problem ◽  
Random Variables ◽  
Total Positivity ◽  
Bivariate Distribution ◽  
Discrete Distributions ◽  
Optimal Solutions ◽  
Totally Positive ◽  
Maximal Correlation ◽  
Optimal Coupling

An optimal coupling is a bivariate distribution with specified marginals achieving maximal correlation. We show that optimal couplings are totally positive and, in fact, satisfy a strictly stronger condition we call the nonintersection property. For discrete distributions we illustrate the equivalence between optimal coupling and a certain transportation problem. Specifically, the optimal solutions of greedily-solvable transportation problems are totally positive, and even nonintersecting, through a rearrangement of matrix entries that results in a Monge sequence. In coupling continuous random variables or random vectors, we exploit a characterization of optimal couplings in terms of subgradients of a closed convex function to establish a generalization of the nonintersection property. We argue that nonintersection is not only stronger than total positivity, it is the more natural concept for the singular distributions that arise in coupling continuous random variables.

scholarly journals Hierarchical Non-Negative Matrix Factorization Using Clinical Information for Microbial Communities.

2021 ◽  
Author(s):  
Ko Abe ◽  
Masaaki Hirayama ◽  
Kinji Ohno ◽  
Teppei Shimamura
Keyword(s):  
Environmental Factors ◽  
Microbial Communities ◽  
Matrix Factorization ◽  
Semantic Analysis ◽  
Human Microbiome ◽  
Nonnegative Matrix ◽  
Clinical Information ◽  
Efficient Procedure ◽  
Estimated Parameters ◽  
The Relationship

Abstract Background: The human microbiome forms very complex communities that consist of hundreds to thousands of different microorganisms that not only affect the host, but also participate in disease processes. Several state-of-the-art methods have been proposed for learning the structure of microbial communities and to investigate the relationship between microorganisms and host environmental factors. However, these methods were mainly designed to model and analyze single microbial communities that do not interact with or depend on other communities. Such methods therefore cannot comprehend the properties between interdependent systems in communities that affect host behavior and disease processes. Results: We introduce a novel hierarchical Bayesian framework, called BALSAMICO (BAyesian Latent Semantic Analysis of MIcrobial COmmunities), which uses microbial metagenome data to discover the underlying microbial community structures and the associations between microbiota and their environmental factors. BALSAMICO models mixtures of communities in the framework of nonnegative matrix factorization, taking into account environmental factors. This method first proposes an efficient procedure for estimating parameters. A simulation then evaluates the accuracy of the estimated parameters. Finally, the method is used to analyze clinical data. In this analysis, we successfully detected bacteria related to colorectal cancer. These results show that the method not only accurately estimates the parameters needed to analyze the connections between communities of microbiota and their environments, but also allows for the effective detection of these communities in real-world circumstances.

scholarly journals Comprehensive Analysis of the Systemic Transcriptomic Alternations and Inflammatory Response during the Occurrence and Progress of COVID-19

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Shaocong Mo ◽  
Leijie Dai ◽  
Yulin Wang ◽  
Biao Song ◽  
Zongcheng Yang ◽  
...  
Keyword(s):  
Inflammatory Response ◽  
Global Economy ◽  
Immune Cell ◽  
Nonnegative Matrix ◽  
Neutrophil Degranulation ◽  
Linear Correlation Analysis ◽  
The Relationship ◽  
Coexpression Network Analysis

The pandemic of the coronavirus disease 2019 (COVID-19) has posed huge threats to healthcare systems and the global economy. However, the host response towards COVID-19 on the molecular and cellular levels still lacks full understanding and effective therapies are in urgent need. Here, we integrate three datasets, GSE152641, GSE161777, and GSE157103. Compared to healthy people, 314 differentially expressed genes were identified, which were mostly involved in neutrophil degranulation and cell division. The protein-protein network was established and two significant subsets were filtered by MCODE: ssGSEA and CIBERSORT, which comprehensively revealed the alternation of immune cell abundance. Weighted gene coexpression network analysis (WGCNA) as well as GO and KEGG analyses unveiled the role of neutrophils and T cells during the progress of the disease. Based on the hospital-free days after 45 days of follow-up and statistical methods such as nonnegative matrix factorization (NMF), submap, and linear correlation analysis, 31 genes were regarded as the signature of the peripheral blood of COVID-19. Various immune cells were identified to be related to the prognosis of the patients. Drugs were predicted for the genes in the signature by DGIdb. Overall, our study comprehensively revealed the relationship between the inflammatory response and the disease course, which provided strategies for the treatment of COVID-19.

scholarly journals TOTAL POSITIVITY IN SPRINGER FIBRES

2020 ◽  
Author(s):  
G Lusztig
Keyword(s):  
Reductive Group ◽  
Total Positivity ◽  
Flag Manifold ◽  
Cell Decomposition ◽  
Unipotent Element ◽  
Positive Part ◽  
Totally Positive ◽  
Natural Cell

Abstract Let u be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at u with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold.

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