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

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.

scholarly journals A dynamical systems view of network centrality

2014 ◽  
Vol 470 (2165) ◽  
pp. 20130835 ◽  
Author(s):  
Peter Grindrod ◽  
Desmond J. Higham
Keyword(s):  
Dynamical Systems ◽  
Large Scale ◽  
Dynamic Networks ◽  
Network Evolution ◽  
Mathematical Framework ◽  
The Novel ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Systems View ◽  
Logarithm Function

To gain insights about dynamic networks, the dominant paradigm is to study discrete snapshots , or timeslices , as the interactions evolve. Here, we develop and test a new mathematical framework where network evolution is handled over continuous time, giving an elegant dynamical systems representation for the important concept of node centrality. The resulting system allows us to track the relative influence of each individual. This new setting is natural in many digital applications, offering both conceptual and computational advantages. The novel differential equations approach is convenient for modelling and analysis of network evolution and gives rise to an interesting application of the matrix logarithm function. From a computational perspective, it avoids the awkward up-front compromises between accuracy, efficiency and redundancy required in the prevalent discrete-time setting. Instead, we can rely on state-of-the-art ODE software, where discretization takes place adaptively in response to the prevailing system dynamics. The new centrality system generalizes the widely used Katz measure, and allows us to identify and track, at any resolution, the most influential nodes in terms of broadcasting and receiving information through time-dependent links. In addition to the classical static network notion of attenuation across edges, the new ODE also allows for attenuation over time, as information becomes stale. This allows ‘running measures’ to be computed, so that networks can be monitored in real time over arbitrarily long intervals. With regard to computational efficiency, we explain why it is cheaper to track good receivers of information than good broadcasters. An important consequence is that the overall broadcast activity in the network can also be monitored efficiently. We use two synthetic examples to validate the relevance of the new measures. We then illustrate the ideas on a large-scale voice call network, where key features are discovered that are not evident from snapshots or aggregates.

scholarly journals Split-and-Combine Singular Value Decomposition for Large-Scale Matrix

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jengnan Tzeng
Keyword(s):  
Singular Value Decomposition ◽  
Large Scale ◽  
Semantic Analysis ◽  
Computational Cost ◽  
Matrix Decomposition ◽  
Singular Value ◽  
Fast Method ◽  
The Matrix ◽  
Scale Matrix ◽  
Value Decomposition

The singular value decomposition (SVD) is a fundamental matrix decomposition in linear algebra. It is widely applied in many modern techniques, for example, high- dimensional data visualization, dimension reduction, data mining, latent semantic analysis, and so forth. Although the SVD plays an essential role in these fields, its apparent weakness is the order three computational cost. This order three computational cost makes many modern applications infeasible, especially when the scale of the data is huge and growing. Therefore, it is imperative to develop a fast SVD method in modern era. If the rank of matrix is much smaller than the matrix size, there are already some fast SVD approaches. In this paper, we focus on this case but with the additional condition that the data is considerably huge to be stored as a matrix form. We will demonstrate that this fast SVD result is sufficiently accurate, and most importantly it can be derived immediately. Using this fast method, many infeasible modern techniques based on the SVD will become viable.

scholarly journals Publicly Verifiable Outsourcing Computation for QR Decomposition Based on Blockchain

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Huimin Wang ◽  
Dong Zheng ◽  
Qinglan Zhao
Keyword(s):  
Large Scale ◽  
Qr Decomposition ◽  
Limited Resources ◽  
The Public ◽  
Security Challenges ◽  
The Matrix ◽  
Cdh Assumption ◽  
Computation Scheme ◽  
Scale Matrix ◽  
Public Verification

In the Big Data Era, outsourcing computation has become increasingly significant as it supplies computation resources for clients with limited resources. However, there are still many security challenges such as payment fairness, privacy protection, and verification. In this paper, we propose a secure publicly verifiable outsourcing computation scheme for the large-scale matrix QR decomposition. In the proposed scheme, client can pay for outsourcing services through blockchain-based payment system which achieves the payment fairness. Moreover, to protect privacy, both permutation matrix and block diagonal matrix are applied in encryption process. Meanwhile, to achieve the public verification, the computational complexity is reduced by using the matrix digest technology. It is worth mentioning that our scheme is provable and secure under the co-CDH assumption.

scholarly journals Carbon Nanotubes and Polydopamine Modified Poly(dimethylsiloxane) Sponges for Efficient Oil–Water Separation

Materials ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2431
Author(s):  
Wen Zhang ◽  
Juanjuan Wang ◽  
Xue Han ◽  
Lele Li ◽  
Enping Liu ◽  
...  
Keyword(s):  
Carbon Nanotubes ◽  
Large Scale ◽  
Absorption Capacity ◽  
Potential Candidate ◽  
Hard Template ◽  
Water Separation ◽  
Industrial Oil ◽  
The Matrix ◽  
Oil Water Separation ◽  
Oil Water

In this paper, effective separation of oil from both immiscible oil–water mixtures and oil-in-water (O/W) emulsions are achieved by using poly(dimethylsiloxane)-based (PDMS-based) composite sponges. A modified hard template method using citric acid monohydrate as the hard template and dissolving it in ethanol is proposed to prepare PDMS sponge composited with carbon nanotubes (CNTs) both in the matrix and the surface. The introduction of CNTs endows the composite sponge with enhanced comprehensive properties including hydrophobicity, absorption capacity, and mechanical strength than the pure PDMS. We demonstrate the successful application of CNT-PDMS composite in efficient removal of oil from immiscible oil–water mixtures within not only a bath absorption, but also continuous separation for both static and turbulent flow conditions. This notable characteristic of the CNT-PDMS sponge enables it as a potential candidate for large-scale industrial oil–water separation. Furthermore, a polydopamine (PDA) modified CNT-PDMS is developed here, which firstly realizes the separation of O/W emulsion without continuous squeezing of the sponge. The combined superhydrophilic and superoleophilic property of PDA/CNT-PDMS is assumed to be critical in the spontaneously demulsification process.

scholarly journals Experimental Research on Bond Behaviour of Fabric Reinforced Cementitious Matrix Composites for Retrofitting Masonry Walls

2021 ◽  
Vol 15 (1) ◽  
Author(s):  
Fayu Wang ◽  
Nicholas Kyriakides ◽  
Christis Chrysostomou ◽  
Eleftherios Eleftheriou ◽  
Renos Votsis ◽  
...  
Keyword(s):  
Large Scale ◽  
Tensile Tests ◽  
Seismic Resistance ◽  
Matrix Composites ◽  
Bond Behaviour ◽  
Cementitious Matrix ◽  
The Matrix ◽  
Rc Frame Buildings ◽  
Direct Tensile ◽  
Fabric Reinforced Cementitious Matrix

AbstractFabric reinforced cementitious matrix (FRCM) composites, also known as textile reinforced mortars (TRM), an inorganic matrix constituting fibre fabrics and cement-based mortar, are becoming a widely used composite material in Europe for upgrading the seismic resistance of existing reinforced concrete (RC) frame buildings. One way of providing seismic resistance upgrading is through the application of the proposed FRCM system on existing masonry infill walls to increase their stiffness and integrity. To examine the effectiveness of this application, the bond characteristics achieved between (a) the matrix and the masonry substrate and (b) the fabric and the matrix need to be determined. A series of experiments including 23 material performance tests, 15 direct tensile tests of dry fabric and composites, and 30 shear bond tests between the matrix and brick masonry, were carried out to investigate the fabric-to-matrix and matrix-to-substrate bond behaviour. In addition, different arrangements of extruded polystyrene (XPS) plates were applied to the FRCM to test the shear bond capacity of this insulation system when used on a large-scale wall.

Improved square-root cubature information filter

2016 ◽  
Vol 39 (4) ◽  
pp. 579-588 ◽  
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.

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

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.

Large Scale Matrix Problems.

1983 ◽  
Vol 40 (161) ◽  
pp. 413
Author(s):  
Ake Bjorck ◽  
Robert J. Plemmons ◽  
Hans Schneider
Keyword(s):  
Large Scale ◽  
Matrix Problems ◽  
Scale Matrix
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