Abstract
In this paper we discuss the notion of singular vector tuples of a complex-valued \(d\)-mode tensor of dimension \(m_1\times \cdots \times m_d\). We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the corresponding Segre product. We give the formula for the number of singular vector tuples. We show similar results for tensors with partial symmetry. We give analogous results for the homogeneous pencil eigenvalue problem for cubic tensors, i.e., \(m_1=\cdots =m_d\). We show the uniqueness of best approximations for almost all real tensors in the following cases: rank-one approximation; rank-one approximation for partially symmetric tensors (this approximation is also partially symmetric); rank-\((r_1,\ldots ,r_d)\) approximation for \(d\)-mode tensors.
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References
S. Banach, Über homogene Polynome in (\(L^2\)), Studi. Math. 7 (1938), 36–44.
E. Carlini and J. Kleppe, Ranks derived from multilinear maps, J. Pure Appl. Algebra 215 (2011), 1999–2004.
D. Cartwright and B. Sturmfels, The number of eigenvectors of a tensor, Linear Algebra Appl. 438 (2013), no. 2, 942–952.
B. Chen, S. He, Z. Li and S. Zhang, Maximum block improvement and polynomial optimization, SIAM J. Optim. 22 (2012), 87–107.
S. S. Chern, Characteristic classes of Hermitian Manifolds, Ann. Math. 47 (1946), 85–121.
L. de Lathauwer, B. de Moor and J. Vandewalle, On the best rank-1 and rank-\((R_1,\ldots , R_N)\) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl. 21 (2000), 1324–1342.
S. Friedland, Best rank one approximation of real symmetric tensors can be chosen symmetric, Front. Math. China 8 (2013), 19 40.
W. Fulton, Intersection Theory, Springer, Berlin (1984).
I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston, 1994.
G. H. Golub and C. F. Van Loan, Matrix Computations, John Hopkins University Press, Baltimore, MD, 3rd Ed., (1996).
P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley (1978).
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer, New York (1977).
C. J. Hillar and L.-H. Lim. Most tensor problems are NP hard, J. ACM 60 (2013), no. 6, Art. 45, 39 pp.
F. Hirzebruch, Topological Methods in Algebraic Geometry, Grundlehren der math. Wissenschaften, vol. 131, Springer (1966).
S. Kobayashi, Differential Geometry of Complex Vector Bundles, Princeton University Press (1987).
L.-H. Lim. Singular values and eigenvalues of tensors: a variational approach. Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP ’05), vol. 1 (2005), 129–132.
L. Lyusternik and L. Shnirel’man, Topological methods in variational problems and their application to the differential geometry of surfaces. (Russian) Uspehi Matem. Nauk (N.S.) 2 (1947), no. 1(17), 166–217.
C. Massri, Algorithm to find a maximum of a multilinear map over a product of spheres, J. Approx. Theory 166 (2013), 19–41.
G. Ni, L. Qi, F. Wang and Y. Wang, The degree of the \(E\)-characteristic polynomial of an even order tensor, J. Math. Anal. Appl. 329 (2007), no. 2, 1218–1229.
L. Oeding and G. Ottaviani, Eigenvectors of tensors and algorithms for Waring decomposition, J. Symb. Comput. 54 (2013), 9–35.
L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput. 40 (2005) 1302–1324.
L. Qi, Eigenvalues and invariants of tensors, J. Math. Anal. Appl. 325 (2007) 1363–1377.
X. Zhang, C. Ling and L. Qi, The best rank-1 approximation of a symmetric tensor and related spherical optimization problems, SIAM J. Matrix Anal. Appl. 33 (2012) 806–821.
Acknowledgments
Shmuel Friedland was supported by National Science Foundation Grant DMS-1216393. Giorgio Ottaviani is member of INDAM.
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Communicated by Peter Buergisser.
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Friedland, S., Ottaviani, G. The Number of Singular Vector Tuples and Uniqueness of Best Rank-One Approximation of Tensors. Found Comput Math 14, 1209–1242 (2014). https://doi.org/10.1007/s10208-014-9194-z
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DOI: https://doi.org/10.1007/s10208-014-9194-z
Keywords
- Singular vector tuples
- Vector bundles
- Chern classes
- Partially symmetric tensors
- Homogeneous pencil eigenvalue problem for cubic tensors
- Singular value decomposition
- Best rank-one approximation
- Best rank-\((r_1 , \ldots , r_d)\) approximation