Abstract
In the tensor space \({{\mathrm {Sym}}}^d \mathbb {R}^2\) of binary forms we study the best rank k approximation problem. The critical points of the best rank 1 approximation problem are the eigenvectors and it is known that they span a hyperplane. We prove that the critical points of the best rank k approximation problem lie in the same hyperplane. As a consequence, every binary form may be written as linear combination of its critical rank 1 tensors, which extends the Spectral Theorem from quadratic forms to binary forms of any degree. In the same vein, also the best rank k approximation may be written as a linear combination of the critical rank 1 tensors, which extends the Eckart–Young theorem from matrices to binary forms.
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References
Abo, H., Eklund, D., Kahle, T., Peterson, C.: Eigenschemes and the Jordan canonical form. Linear Algebra Appl. 496, 121–151 (2016)
Abo, H., Seigal, A., Sturmfels, B.: Eigenconfigurations of tensors. Algebr. Geom. Methods Discrete Math. Contemp. Math. 685, 1–25 (2017)
Draisma, J., Horobeţ, E., Ottaviani, G., Sturmfels, B., Thomas, R.: The Euclidean distance degree of an algebraic variety. Found. Comput. Math. 16(1), 99–149 (2016)
Draisma, J., Ottaviani, G., Tocino, A.: Best rank \(k\) approximation for tensors. In preparation (2017)
Friedland, S., Tammali, V.: Low-rank approximation of tensors, numerical algebra, matrix theory, differential-algebraic equations and control theory, pp. 377–411. Springer, Cham (2015)
Lee, H., Sturmfels, B.: Duality of multiple root loci. J. Algebra 446, 499–526 (2016)
Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach. In: Proc. IEEE Internat. Workshop on Comput. Advances in Multi-sensor Adaptive Processing, CAMSAP, pp. 129–132 (2005)
Maccioni, M.: The number of real eigenvectors of a real polynomial. Boll. Unione Mat. Ital. (2017). doi:10.1007/s40574-016-0112-y
Ottaviani, G., Paoletti, R.: A geometric perspective on the singular value decomposition. Rend. Istit. Mat. Univ. Trieste 47, 107–125 (2015)
Ottaviani, G., Spaenlehauer, P.J., Sturmfels, B.: Exact solutions in structured low-rank approximation. SIAM J. Matrix Anal. Appl. 35(4), 1521–1542 (2014)
Reznick, B.: Sums of Even Powers of Real Linear Forms, vol. 463, p. 96. Mem. Am. Math. Soc., Providence (1992)
Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)
Seigal, A., Sturmfels, B.: Real rank two geometry. J. Algebra 484, 310–333 (2017)
Sturmfels, B.: Tensors and their eigenvectors. Not. Am. Math. Soc. 63, 604–606 (2016)
Acknowledgements
Giorgio Ottaviani is member of INDAM-GNSAGA. This paper has been partially supported by the Strategic Project “Azioni di Gruppi su varietà e tensori” of the University of Florence.
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Ottaviani, G., Tocino, A. Best rank k approximation for binary forms. Collect. Math. 69, 163–171 (2018). https://doi.org/10.1007/s13348-017-0206-6
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DOI: https://doi.org/10.1007/s13348-017-0206-6