Shifted Power Method for Computing Tensor Eigenpairs

By Tamara G. Kolda, Jackson R. Mayo. SIAM Journal on Matrix Analysis and Applications, 2010

384 citations recorded by the source corpus.

Recent work on eigenvalues and eigenvectors for tensors of order $m \ge 3$ has been motivated by applications in blind source separation, magnetic resonance imaging, molecular conformation, and more. In this paper, we consider methods for computing real symmetric-tensor eigenpairs of the form $\boldsymbol{\mathscr{A}}\mathbf{x}^{m-1} = \lambda \mathbf{x}$ subject to $\|\mathbf{x}\|=1$, which is closely related to optimal rank-1 approximation of a symmetric tensor. Our contribution is a shifted symmetric higher-order power method (SS-HOPM), which we show is guaranteed to converge to a tensor eigenpair. SS-HOPM can be viewed as a generalization of the power iteration method for matrices or of the symmetric higher-order power method. Additionally, using fixed point analysis, we can characterize exactly which eigenpairs can and cannot be found by the method. Numerical examples are presented, including examples from an extension of the method to finding complex eigenpairs.

Subfields: Computational Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/f7ef8d06770db848a0f7f89bc5ae5fe3f99c7d4c

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Citations in the corpus

Nonsingular $H$-tensor and its criteria (2016, background): “Hence, tensor analysis and computing have received much attention of researchers in recent decade and the research branches include spectral hypergraph theory [13, 29], best low-rank approximation to tensors [15, 17, 37], positive definiteness identification of even order homogenous polynomials [12,”
Tensors for Data Mining and Data Fusion (2016, methodology): “This can be expressed as a tensor eigenvalue problem, and the method of Kolda and Mayo [2011] can be used to solve it.”
Z-EIGENVALUE INCLUSION THEOREMS FOR TENSORS (2016, methodology): “Some effective algorithms for finding Z-eigenvalue and the corresponding eigenvector of tensors have been implemented [3, 6, 10, 16].”
A tensor approach to learning mixed membership community models (2013, methodology): “Finally, we note that stabilization, as proposed by Kolda and Mayo (2011) for general tensor eigen-decomposition (as opposed to orthogonal decomposition in this paper), can be effective in improving convergence, especially on real data, and we defer its detailed analysis to future work.”
An S-type eigenvalue localization set for tensors (2015, background): “Introduction Eigenvalue problems of tensors have become an important topic of study in numerical multilinear algebra, and they have a wide range of practical applications; see [2, 3, 4, 5, 7, 8, 9, 10, 11, 17, 18, 19, 21, 22, 24, 25].”
Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging (2013, background): “Eigenvalue decomposition for tensors means different things to different researchers (see [12, 16, 18, 19], for example).”
Jacobi Algorithm for the Best Low Multilinear Rank Approximation of Symmetric Tensors (2013, background): “Fixes for this problem have been proposed in the literature for the case of rank-(1, 1, 1) approximation [31, 42, 33].”
A new eigenvalue inclusion set for tensors and its applications (2015, background): “Recently, tensor eigenvalues and eigenvectors have received much attention in the literature [1,8,9,11,15,18–22,25].”
Tensor decompositions for learning latent variable models (2012, background): “The practical value of addressing these oscillations and perhaps stabilizing the algorithm is an interesting direction for future research (Kolda and Mayo, 2011).”