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Computing tensor Z-eigenvectors with dynamical systems
[article]
2019
arXiv
pre-print
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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]
2022
arXiv
pre-print
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2021 pre-print arXiv:2107.11438v2 fatcat:gqbo66fz2fdw5by4lvsrponqwmOther Versions
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
2017
Chaos
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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
2009
Physical Review Letters
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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
2019
SIAM Journal on Mathematics of Data Science
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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
2012
New Journal of Physics
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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]
2019
arXiv
pre-print
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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
1992
Perception & Psychophysics
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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]
2020
arXiv
pre-print
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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]
2006
Mathematics and Visualization
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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
2014
Physical Review B
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2014 pre-print arXiv:1401.6762v1 fatcat:3rjsxfbq4rh4lc3ffwhcbkaw2m
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
1991
Geophysical Journal International
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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]
2012
Mathematics and Visualization
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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
2015
Intelligent Data Analysis
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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
2002
Liquid crystals (Print)
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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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