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Computing tensor Z-eigenvectors with dynamical systems [article]

Austin R. Benson, David F. Gleich
2019 arXiv   pre-print
We present a new framework for computing Z-eigenvectors of general tensors based on numerically integrating a dynamical system that can only converge to a Z-eigenvector.  ...  Our motivation comes from our recent research on spacey random walks, where the long-term dynamics of a stochastic process are governed by a dynamical system that must converge to a Z-eigenvector of a  ...  A dynamical systems framework for computing Z-eigenvectors. Observation 1.1 provides a new perspective on the tensor Z-eigenvector problem.  ... 
arXiv:1805.00903v3 fatcat:attkam4jifd67h4k5kibyp6hgq

Explicit Solutions and Stability Properties of Homogeneous Polynomial Dynamical Systems [article]

Can Chen
2022 arXiv   pre-print
Z-eigenvalues and Z-eigenvectors.  ...  In this paper, we provide a system theoretic treatment of certain continuous-time homogeneous polynomial dynamical systems (HPDS) via tensor algebra.  ...  Similarly, in this paper, we will exploit tensor orthogonal decomposition with Z-eigenvalues and Z-eigenvectors to summarize the explicit polynomial dynamical systems with constant inputs in subsection  ... 
arXiv:2107.11438v3 fatcat:uajjm7oh6jesxguouqzynfjnue

Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents

Hessam Babaee, Mohamad Farazmand, George Haller, Themistoklis P. Sapsis
2017 Chaos  
High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have finite-time duration.  ...  In high-dimensional systems, the computational cost of the reduced-order method is orders of magnitude lower than the full FTLE computation.  ...  Let n i ðt; t 0 ; z 0 Þ and g i ðt; t 0 ; z 0 Þ ði ¼ 1; 2; …; nÞ denote the eigenvectors of the right and left Cauchy-Green strain tensors, respectively, with corresponding eigenvalues k 1 ðt; t 0 ; z  ... 
doi:10.1063/1.4984627 pmid:28679218 fatcat:3ogi6mskdrfkhjuvomv4js2csq

Matrix Product States for Dynamical Simulation of Infinite Chains

M. C. Bañuls, M. B. Hastings, F. Verstraete, J. I. Cirac
2009 Physical Review Letters  
By folding the network in the time direction prior to contraction, time dependent expectation values and dynamic correlation functions can be computed after much longer evolution time than with any previous  ...  We propose a new method for computing the ground state properties and the time evolution of infinite chains based on a transverse contraction of the tensor network.  ...  The standard way of computing dynamical quantities with the MPS formalism starts with a state that is (exactly) described by a MPS (1).  ... 
doi:10.1103/physrevlett.102.240603 pmid:19658990 fatcat:e7wtf5cmmzdm5ikyvv4djo7zmi

Three Hypergraph Eigenvector Centralities

Austin R. Benson
2019 SIAM Journal on Mathematics of Data Science  
Using recently established Perron--Frobenius theory for tensors, we develop three tensor eigenvectors centralities for hypergraphs, each with different interpretations.  ...  Eigenvector centrality is a standard network analysis tool for determining the importance of (or ranking of) entities in a connected system that is represented by a graph.  ...  Recent work by the author develops a method to compute Z-eigenpairs using dynamical systems, which can scale to large tensors and also compute unstable eigenvectors , albeit without theoretical guarantees  ... 
doi:10.1137/18m1203031 fatcat:65txakz4r5c2xgujgi5jmskotm

Tensor network techniques for the computation of dynamical observables in one-dimensional quantum spin systems

Alexander Müller-Hermes, J Ignacio Cirac, Mari Carmen Bañuls
2012 New Journal of Physics  
We benchmark the accomplishments of this technique with respect to alternative strategies using Ising Hamiltonians with transverse and parallel fields, as well as XY models.  ...  Lett. 102, 240603 (2009)] to simulate the dynamics of infinite quantum spin chains, and relate its performance to the kind of entanglement produced under the evolution of product states.  ...  Thirdly, we describe various dynamical observables that can be computed with the transverse techniques, and benchmark their performances for evolution after a global quench and for computing dynamical  ... 
doi:10.1088/1367-2630/14/7/075003 fatcat:3e4gcgrogjatdn5kyqqcduygrq

Three hypergraph eigenvector centralities [article]

Austin R. Benson
2019 arXiv   pre-print
Using recently established Perron-Frobenius theory for tensors, we develop three tensor eigenvectors centralities for hypergraphs, each with different interpretations.  ...  Eigenvector centrality is a standard network analysis tool for determining the importance of (or ranking of) entities in a connected system that is represented by a graph.  ...  Recent work by the author develops a method to compute Z-eigenpairs using dynamical systems, which can scale to large tensors and also compute unstable eigenvectors [Benson and Gleich, 2018] , albeit  ... 
arXiv:1807.09644v3 fatcat:hpf5gohbl5exbfb36vmtmvqkhq

Eigenvectors of the inertia tensor and perceiving the orientation of a hand-held object by dynamic touch

Christopher C. Pagano, M. T. Turvey
1992 Perception & Psychophysics  
Perceived orientation was found to be dependent on the eigenvectors of the object's inertia tensor, computed about the point of rotation in the wrist, rather than on its spatial orientation.  ...  The results underscore the significance of the inertia tensor to understanding the perception of spatial properties by dynamic touch.  ...  This invariant form is with respect to the principal axes or eigenvectors off-the only nonarbitrary coordinate system at O.  ... 
doi:10.3758/bf03211699 pmid:1287567 fatcat:rz5ac7ney5gppggqhkcedzms3u

Time-dependent variational principle of mixed matrix product states in the thermodynamic limit [article]

Yantao Wu
2020 arXiv   pre-print
It is inversion-free and very simple to adapt from an existing TDVP code for finite systems. The importance of working in the projective Hilbert space is highlighted.  ...  We follow [15] to compute the quench dynamics.  ...  The quenching Hamiltonian is δĤ =σ z i0 in both cases. The computation is done with δt = 0.005 and D = 20.  ... 
arXiv:2007.15035v2 fatcat:zrylr4avujdphgdn64xu456faa

Locating Closed Hyperstreamlines in Second Order Tensor Fields [chapter]

Thomas Wischgoll, Joerg Meyer
2006 Mathematics and Visualization  
Topology-based methods that investigate the eigenvector fields of second order tensor fields have gained increasing interest in recent years.  ...  The analysis and visualization of tensor fields is an advancing area in scientific visualization.  ...  and Computer Science Department of the University of California, Irvine.  ... 
doi:10.1007/3-540-31272-2_15 fatcat:phifzche75c5nkz2wwrzde3aku

Long-time dynamics of quantum chains: Transfer-matrix renormalization group and entanglement of the maximal eigenvector

Yu-Kun Huang, Pochung Chen, Ying-Jer Kao, Tao Xiang
2014 Physical Review B  
Our method exhibits a competitive accuracy with a much cheaper computational cost in comparison with two recent proposed methods for long-time dynamics based on a folding algorithm [Phys. Rev.  ...  By using a different quantum-to-classical mapping from the Trotter-Suzuki decomposition, we identify the entanglement structure of the maximal eigenvectors for the associated quantum transfer matrix.  ...  to address real-time dynamics [2] and the computation of thermodynamic quantities [3] .  ... 
doi:10.1103/physrevb.89.201102 fatcat:bs75ucga2bbcxbybufspivuriq

On covariances of eigenvalues and eigenvectors of second-rank symmetric tensors

Tomás Soler, Boudewijn H. W. Gelder
1991 Geophysical Journal International  
This work expands the conventional methodology by introducing equations to compute the covariance matrices of eigenvalues and eigenvectors of second-rank 3-D symmetric tensors in terms of their six distinct  ...  New analytical expressions derived herein are general and should be of interest to anyone concerned with the accuracy of the computed orientation of principal axes and their associated principal quantities  ...  INTRODUCTION In mechanics, dynamics, statistics, etc. symmetric 3-D second-rank tensors play important roles.  ... 
doi:10.1111/j.1365-246x.1991.tb06732.x fatcat:4ceuexhvgnhbjlu7jnir27bnya

Beyond Topology: A Lagrangian Metaphor to Visualize the Structure of 3D Tensor Fields [chapter]

Xavier Tricoche, Mario Hlawitschka, Samer Barakat, Christoph Garth
2012 Mathematics and Visualization  
As a result, we propose an alternative structure characterization strategy for the visual analysis of practical 3D tensor fields, which we demonstrate on several synthetic and computational datasets.  ...  Indeed, one can view eigenvector fields as the local superimposition of two vector fields, from which a bidirectional flow field can be defined.  ...  Acknowledgements This work was supported in part by a gift from Intel Visual Computing initiative.  ... 
doi:10.1007/978-3-642-27343-8_5 fatcat:pfqgqkeyfrgkfpsddeai66oazi

EigenEvent: An algorithm for event detection from complex data streams in syndromic surveillance

Hadi Fanaee-T, João Gama
2015 Intelligent Data Analysis  
The type of data that is being generated via these systems is usually multivariate and seasonal with spatial and temporal dimensions.  ...  Syndromic surveillance systems continuously monitor multiple pre-diagnostic daily streams of indicators from different regions with the aim of early detection of disease outbreaks.  ...  ; and 3) dynamic baseline tensor (our strategy).  ... 
doi:10.3233/ida-150734 fatcat:hibrpp3zpnemtfs5igfqouj75e

A vector grouping algorithm for liquid crystal tensor field visualization

Yang-Ming Zhu, Paul A. Farrell
2002 Liquid crystals (Print)  
Tensor elds are at the heart of many science and engineering disciplines. Many tensor visualization methods separate the tensor into component eigenvectors and visualize those instead.  ...  Eigenvectors are normally ordered according to their eigenvalues: the eigenvectors corresponding to the smallest, median, or largest eigenvalues are in their corresponding groups.  ...  For i = 0 1 N ; 1 j = 0 1 N ; 1 k = 1 2 N ; 1, and for the six permutations of (U V W) a t ( x i y j z k ), compute X <x y z> (U x y z U i j k ) 2 + ( V x y z V i j k ) 2 + ( W x y z W i j k ) 2 where  ... 
doi:10.1080/713935624 fatcat:g6ojh6cdcjew3kfwrfqxg3rfam
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