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Perturbation analysis of third-order tensor eigenvalue problem based on tensor-tensor multiplication [article]

Changxin Mo, Weiyang Ding
2021 arXiv   pre-print
In this paper, we pay our attention to the perturbation analysis subject on tensor eigenvalues under tensor-tensor multiplication sense; and also ϵ-pseudospectra theory for third-order tensors.  ...  The definition of the generalized T-eigenvalue of third-order tensors is given.  ...  Perturbation analysis on third-order tensors In this section, we firstly give a definition of the generalized tensor-eigenvalue of thirdorder tensors.  ... 
arXiv:2108.09502v1 fatcat:yknpylmcgbgy5ai5xa2wxfuxlm

A note on perturbation analysis for T-product based tensor singular values [article]

Yating Zhang, Xiaoxia Guo, Pengpeng Xie
2021 arXiv   pre-print
In this note, we present perturbation analysis for the T-product based tensor singular values defined by Lu et al. First, the Cauchy's interlacing-type theorem for tensor singular values is given.  ...  Finally, we introduce some useful inequalities for the singular values of tensor products and sums.  ...  Based on this tensortensor multiplication, many scholars have contributed to the theoretical results and algebraic analysis [5, 9] .  ... 
arXiv:2109.11194v1 fatcat:hmhfrupmqjepjgypf3jhgxezdu

Using Perturbation Theory to Compute the Morphological Similarity of Diffusion Tensors

R. Bansal, L.H. Staib, Dongrong Xu, A.F. Laine, J. Royal, B.S. Peterson
2008 IEEE Transactions on Medical Imaging  
Our measure presupposes that DTs in a homogeneous region within or across DT images are random perturbations of one another in the presence of noise.  ...  Although comparing tensor elements, rather than PDs, is comparatively more robust to the effects of noise, the individual elements of a given tensor do not directly reflect the diffusion properties of  ...  We use a nondegenerate analysis when each eigenvalue is unique and a degenerate analysis when 2 or 3 eigenvalues of a tensor equal one another.  ... 
doi:10.1109/tmi.2007.912391 pmid:18450533 pmcid:PMC2398733 fatcat:mrmhdukpgbgedbmrpaxrojgpke

A Multilinear Singular Value Decomposition

Lieven De Lathauwer, Bart De Moor, Joos Vandewalle
2000 SIAM Journal on Matrix Analysis and Applications  
There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, first-order perturbation effects  ...  We investigate how tensor symmetries affect the decomposition and propose a multilinear generalization of the symmetric eigenvalue decomposition for pair-wise symmetric tensors.  ...  As pseudodiagonality is a special case of all-orthogonality, analysis procedures may then be based on the HOSVD, up to perturbation effects.  ... 
doi:10.1137/s0895479896305696 fatcat:toefaifrkvdgdgu5sc7sc46n4y

Квантовая интегрируемость операторов Бельтрами — Лапласа проективно эквивалентных метрик произвольных сигнатур

Владимир Сергеевич Матвеев
2020 Чебышевский сборник  
Let us now consider the case when has one real eigenvalue of geometric multiplicity 1, or a pair of nonreal complex-conjugate eigenvalues of geometric multiplicity 1.  ...  Note that is the half of the sum of eigenvalues of , counted with algebraic multiplicities.  ... 
doi:10.22405/2226-8383-2020-21-2-275-289 fatcat:wxxx4utcg5a45p6kxbk7m75vcu

A novel strategy to identify the critical conditions for growth-induced instabilities

A. Javili, P. Steinmann, E. Kuhl
2014 Journal of The Mechanical Behavior of Biomedical Materials  
This kinematic approach towards growth-induced instabilities is based on the multiplicative decomposition of the total deformation gradient into a reversible elastic part and an irreversible growth part  ...  Here we propose a novel strategy for the stability analysis of growing structures motivated by the idea of replacing growth by prestress.  ...  Acknowledgements This study was supported in part by the National Science Foundation CAREER award CMMI 0952021 and INSPIRE grant 1233054 and by the National Institutes of Health Grant U54 GM072970.  ... 
doi:10.1016/j.jmbbm.2013.08.017 pmid:24041754 fatcat:quim555ckvccbpfa4st3p3tzii

Effective Channel Order Determination Algorithm for Convolutive Blind Channel Identification

Senquan Yang, Haifeng Su, Pu Li, Songxi Hu, Jinru Chen
2019 International Journal of Computational Intelligence Systems  
Effective channel order determination Numerical analysis arguments Blind channel identification Higher-order cumulant tensor A B S T R A C T Effective channel order determination is an important problem  ...  The classical techniques are based on information theoretic criteria, which show a great potentiality to estimate the effective channel order.  ...  In addition, an effective method has been proposed based on numerical analysis arguments [20, 21] , which detects the effective channel order using a rank detection method to the single-input/multiple-output  ... 
doi:10.2991/ijcis.d.190819.001 fatcat:bqg6d6366zgermm5eowknnqp24

Numerical multilinear algebra and its applications

Liqun Qi, Wenyu Sun, Yiju Wang
2007 Frontiers of Mathematics in China  
Keywords Numerical multilinear algebra, higher order tensor, tensor decomposition, lower rank approximation of tensor, multi-way data analysis MSC 15A18, 15A69, 65F30 *  ...  Numerical multilinear algebra, in which instead of matrices and vectors the higher-order tensors are considered in numerical viewpoint, is a new branch of computational mathematics.  ...  stability and perturbation analysis of tensor computation, and so on.  ... 
doi:10.1007/s11464-007-0031-4 fatcat:hqjo2wdgmbdtvixgjpysypzmw4

Perturbation Theory and Optimality Conditions for the Best Multilinear Rank Approximation of a Tensor

Lars Eldén, Berkant Savas
2011 SIAM Journal on Matrix Analysis and Applications  
The problem of computing the best rank-(p, q, r) approximation of a third order tensor is considered.  ...  First the problem is reformulated as a maximization problem on a product of three Grassmann manifolds.  ...  The analysis is based on the differential-geometric framework developed in [12] .  ... 
doi:10.1137/110823298 fatcat:mn7wbehlabcujesotddjzqpnne

Visualization of Geologic Stress Perturbations Using Mohr Diagrams

P. Crossno, D.H. Rogers, R.M. Brannon, D. Coblentz, J.T. Fredrich
2005 IEEE Transactions on Visualization and Computer Graphics  
as a new tensor glyph for dynamically exploring tensor variables within three-dimensional finite element models.  ...  We present an adaptation of the Mohr diagram, a traditional paper-and-pencil graphical method long used by the material mechanics community for estimating coordinate transformations for stress tensors,  ...  The Department of Energy (DOE) Mathematics, Information, and Computer Science Office funded the visualization portion of this work.  ... 
doi:10.1109/tvcg.2005.86 pmid:16144248 fatcat:hnslvfmsnzbhrhmfrnfdincmpm

Diffusion tensor smoothing through weighted Karcher means

Owen Carmichael, Jun Chen, Debashis Paul, Jie Peng
2013 Electronic Journal of Statistics  
Leading approaches to this problem amount to estimation of weighted Karcher means of Diffusion tensors within spatial neighborhoods, under various metrics imposed on the space of tensors.  ...  In this paper, we combine theoretical analysis, empirical analysis of simulated DTI data, and empirical analysis of real DTI scans to compare the noise removal performance of three kernel-based DTI smoothers  ...  We assume that the underlying true tensors D 0 (ω)'s are perturbed versions of the target tensor D 0 , where the perturbation is additive on the logarithm of the eigenvalues of D 0 , but does not alter  ... 
doi:10.1214/13-ejs825 pmid:25419264 pmcid:PMC4239671 fatcat:k2i4lzlmenhankk6em2wc6yr5y

A Robust Spectral Algorithm for Overcomplete Tensor Decomposition [article]

Samuel B. Hopkins, Tselil Schramm, Jonathan Shi
2022 arXiv   pre-print
Our algorithm is robust to adversarial perturbations of bounded spectral norm.  ...  of measure 0) tensors over (ℝ^d)^⊗ 4 with rank n ≤ d^2.  ...  Acknowledgements We thank David Steurer for many helpful conversations regarding the technical content and presentation of this work.  ... 
arXiv:2203.02790v1 fatcat:22ibxdkx6rh7fn2mkuvlbu4gf4

Visualization of Tensor Fields in Mechanics

Chiara Hergl, Christian Blecha, Vanessa Kretzschmar, Felix Raith, Fabian Günther, Markus Stommel, Jochen Jankowai, Ingrid Hotz, Thomas Nagel, Gerik Scheuermann
2021 Computer graphics forum (Print)  
This survey aims to provide an overview of recent research results with a strong application-oriented focus, targeting applications based on continuum mechanics, namely the fields of structural, bio-,  ...  While tensors encode such complex information mathematically precisely, the semantic interpretation of a tensor is challenging.  ...  [KRZ*19] provide a novel multiscale topological analysis framework for asymmetric tensor fields on surfaces based on these concepts of eigenvalue and eigenvector graphs.  ... 
doi:10.1111/cgf.14209 fatcat:m6grkhl5pndpbbq7zxxgzj3snq

Page 496 of Mathematical Reviews Vol. , Issue 2002A [page]

2002 Mathematical Reviews  
The algorithm can be modified when multiple base eigenvalues arise, although the success of the modification depends on the symmetry properties of the base eigenfunctions.” 2002a:65165 65N22 65F10 65N55  ...  Solutions of the matrix inverse problem exist provided that the reciprocals of the prescribed eigenvalues are close to the reciprocals of the simple eigenvalues of the base problem with p = 1.  ... 

Degenerate 3D Tensors [chapter]

Xiaoqiang Zheng, Xavier Tricoche, Alex Pang
2006 Mathematics and Visualization  
Topological analysis of 3D tensor fields starts with the identification of degeneracies in the tensor field.  ...  This method is based on a description of a tensor having an isotropic spherical component and a linear or planar component. As such, we refer to this formulation as the geometric approach.  ...  Therefore, a proper definition or an extraction algorithm based on both eigenvalues and eigenvectors could provide more insight into this problem.  ... 
doi:10.1007/3-540-31272-2_14 fatcat:x5ocewzy4jeqhcrcpqjw6oe3te
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