On Degenerate Doubly Nonnegative Projection Problems

The doubly nonnegative (DNN) cone, being the set of all positive semidefinite matrices whose elements are nonnegative, is a popular approximation of the computationally intractable completely positive cone. The major difficulty for implementing a Newton-type method to compute the projection of a given large-scale matrix onto the DNN cone lies in the possible failure of the constraint nondegeneracy, a generalization of the linear independence constraint qualification for nonlinear programming. Such a failure results in the singularity of the Jacobian of the nonsmooth equation representing the Karush–Kuhn–Tucker optimality condition that prevents the semismooth Newton–conjugate gradient method from solving it with a desirable convergence rate. In this paper, we overcome the aforementioned difficulty by solving a sequence of better conditioned nonsmooth equations generated by the augmented Lagrangian method (ALM) instead of solving one aforementioned singular equation. By leveraging the metric subregularity of the normal cone associated with the positive semidefinite cone, we derive sufficient conditions to ensure the dual quadratic growth condition of the underlying problem, which further leads to the asymptotically superlinear convergence of the proposed ALM. Numerical results on difficult randomly generated instances and from the semidefinite programming library are presented to demonstrate the efficiency of the algorithm for computing the DNN projection to a very high accuracy.

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THE WISHART SHORT RATE MODEL

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024912500562 ◽

2012 ◽

Vol 15 (08) ◽

pp. 1250056 ◽

Cited By ~ 20

Author(s):

ALESSANDRO GNOATTO

Keyword(s):

Stochastic Process ◽

Sufficient Conditions ◽

Positive Semidefinite ◽

Rate Model ◽

Positive Semidefinite Matrices ◽

Yield Curves ◽

Model Driven

We consider a short rate model, driven by a stochastic process on the cone of positive semidefinite matrices. We derive sufficient conditions ensuring that the model replicates normal, inverse or humped yield curves.

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On Randomized Trace Estimates for Indefinite Matrices with an Application to Determinants

Foundations of Computational Mathematics ◽

10.1007/s10208-021-09525-9 ◽

2021 ◽

Author(s):

Alice Cortinovis ◽

Daniel Kressner

Keyword(s):

Large Scale ◽

Gaussian Process Regression ◽

Likelihood Estimation ◽

Random Vectors ◽

Symmetric Positive Definite ◽

Matrix Logarithm ◽

The Matrix ◽

Trace Estimation ◽

Scale Matrix ◽

Existing Result

AbstractRandomized trace estimation is a popular and well-studied technique that approximates the trace of a large-scale matrix B by computing the average of $$x^T Bx$$ x T B x for many samples of a random vector X. Often, B is symmetric positive definite (SPD) but a number of applications give rise to indefinite B. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite B with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite B, reducing the number of required samples by a factor n or even more, where n is the size of B. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher (Found Comput Math, 15(5):1187–1212, 2015) for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of f(B). Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.

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A model for comparing memory strategies for large scale matrix in linear equation and one experimental case

Proceedings Fourth International Conference/Exhibition on High Performance Computing in the Asia-Pacific Region ◽

10.1109/hpc.2000.843598 ◽

2000 ◽

Author(s):

Yang Zhenxiao ◽

Zhou Haichang ◽

Ba Zhixin

Keyword(s):

Linear Equation ◽

Large Scale ◽

Memory Strategies ◽

Scale Matrix

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Improved square-root cubature information filter

Transactions of the Institute of Measurement and Control ◽

10.1177/0142331215608428 ◽

2016 ◽

Vol 39 (4) ◽

pp. 579-588 ◽

Cited By ~ 10

Author(s):

Yulong Huang ◽

Yonggang Zhang ◽

Ning Li ◽

Lin Zhao

Keyword(s):

Large Scale ◽

Estimation Error ◽

Sampling Interval ◽

Prediction Errors ◽

Square Root ◽

Sigma Point ◽

Theoretical Comparison ◽

Error Covariance ◽

Information Filter ◽

Scale Matrix

In this paper, a theoretical comparison between existing the sigma-point information filter (SPIF) framework and the unscented information filter (UIF) framework is presented. It is shown that the SPIF framework is identical to the sigma-point Kalman filter (SPKF). However, the UIF framework is not identical to the classical SPKF due to the neglect of one-step prediction errors of measurements in the calculation of state estimation error covariance matrix. Thus SPIF framework is more reasonable as compared with UIF framework. According to the theoretical comparison, an improved cubature information filter (CIF) is derived based on the superior SPIF framework. Square-root CIF (SRCIF) is also developed to improve the numerical accuracy and stability of the proposed CIF. The proposed SRCIF is applied to a target tracking problem with large sampling interval and high turn rate, and its performance is compared with the existing SRCIF. The results show that the proposed SRCIF is more reliable and stable as compared with the existing SRCIF. Note that it is impractical for information filters in large-scale applications due to the enormous computational complexity of large-scale matrix inversion, and advanced techniques need to be further considered.

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A trace bound for integer-diagonal positive semidefinite matrices

Special Matrices ◽

10.1515/spma-2020-0002 ◽

2020 ◽

Vol 8 (1) ◽

pp. 14-16

Author(s):

Lon Mitchell

Keyword(s):

Positive Semidefinite ◽

Positive Semidefinite Matrix ◽

Positive Semidefinite Matrices ◽

Semidefinite Matrix

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.

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A perspective on graph theory-based stability analysis of impulsive stochastic recurrent neural networks with time-varying delays

Advances in Difference Equations ◽

10.1186/s13662-019-2443-3 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 10

Author(s):

M. Iswarya ◽

R. Raja ◽

G. Rajchakit ◽

J. Cao ◽

J. Alzabut ◽

...

Keyword(s):

Neural Networks ◽

Graph Theory ◽

Exponential Stability ◽

Recurrent Neural Networks ◽

Lyapunov Functional ◽

Large Scale ◽

Sufficient Conditions ◽

Distributed Delay ◽

Theoretic Approach ◽

Discrete Time Delay

AbstractIn this work, the exponential stability problem of impulsive recurrent neural networks is investigated; discrete time delay, continuously distributed delay and stochastic noise are simultaneously taken into consideration. In order to guarantee the exponential stability of our considered recurrent neural networks, two distinct types of sufficient conditions are derived on the basis of the Lyapunov functional and coefficient of our given system and also to construct a Lyapunov function for a large scale system a novel graph-theoretic approach is considered, which is derived by utilizing the Lyapunov functional as well as graph theory. In this approach a global Lyapunov functional is constructed which is more related to the topological structure of the given system. We present a numerical example and simulation figures to show the effectiveness of our proposed work.

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Bacterial Multicellularity: The Biology of Escherichia coli Building Large-Scale Biofilm Communities

Annual Review of Microbiology ◽

10.1146/annurev-micro-031921-055801 ◽

2021 ◽

Vol 75 (1) ◽

Author(s):

Diego O. Serra ◽

Regine Hengge

Keyword(s):

Escherichia Coli ◽

Large Scale ◽

Second Messengers ◽

Emergent Properties ◽

Annual Review ◽

Publication Date ◽

Growth And Survival ◽

Chemical Gradients ◽

Life Itself ◽

Scale Matrix

Biofilms are a widespread multicellular form of bacterial life. The spatial structure and emergent properties of these communities depend on a polymeric extracellular matrix architecture that is orders of magnitude larger than the cells that build it. Using as a model the wrinkly macrocolony biofilms of Escherichia coli, which contain amyloid curli fibers and phosphoethanolamine (pEtN)-modified cellulose as matrix components, we summarize here the structure, building, and function of this large-scale matrix architecture. Based on different sigma and other transcription factors as well as second messengers, the underlying regulatory network reflects the fundamental trade-off between growth and survival. It controls matrix production spatially in response to long-range chemical gradients, but it also generates distinct patterns of short-range matrix heterogeneity that are crucial for tissue-like elasticity and macroscopic morphogenesis. Overall, these biofilms confer protection and a potential for homeostasis, thereby reducing maintenance energy, which makes multicellularity an emergent property of life itself. Expected final online publication date for the Annual Review of Microbiology, Volume 75 is October 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

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Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

Canadian Journal of Mathematics ◽

10.4153/cjm-2016-022-6 ◽

2017 ◽

Vol 69 (3) ◽

pp. 548-578 ◽

Cited By ~ 2

Author(s):

Michael Hartglass

Keyword(s):

Differential Calculus ◽

Von Neumann Algebras ◽

Sufficient Conditions ◽

Weighted Graph ◽

Positive Cone ◽

Von Neumann ◽

Planar Algebras ◽

C Algebra ◽

C Algebras ◽

Necessary And Sufficient

AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

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