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Decomposition of Low Rank Multi-symmetric Tensor

Conference paper
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Part of the Lecture Notes in Computer Science book series (LNCS, volume 10693)

Abstract

We study the decomposition of a multi-symmetric tensor T as a sum of powers of product of linear forms in correlation with the decomposition of its dual \(T^{*}\) as a weighted sum of evaluations. We use the properties of the associated Artinian Gorenstein Algebra \(A_{\tau }\) to compute the decomposition of its dual \(T^{*}\) which is defined via a formal power series \(\tau \). We use the low rank decomposition of the Hankel operator \(H_{\tau }\) associated to the symbol \(\tau \) into a sum of indecomposable operators of low rank. A basis of \(A_{\tau }\) is chosen such that the multiplication by some variables is possible. We compute the sub-coordinates of the evaluation points and their weights using the eigen-structure of multiplication matrices. The new algorithm that we propose works for small rank. We give a theoretical generalized approach of the method in n dimensional space. We show a numerical example of the decomposition of a multi-linear tensor of rank 3 in 3 dimensional space.

References

  1. 1.
    Anandkumar, A., Ge, R., Hsu, D., Kakade, S.M., Telgarsky, M.: Tensor decompositions for learning latent variable models (A Survey for ALT). In: Chaudhuri, K., Gentile, C., Zilles, S. (eds.) ALT 2015. LNCS, vol. 9355, pp. 19–38. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-24486-0_2 CrossRefGoogle Scholar
  2. 2.
    Bernardi, A., Daleo, N.S., Hauenstein, J.D., Mourrain, B.: Tensor decomposition and homotopy continuation. Differential Geometry and its Applications, August 2017Google Scholar
  3. 3.
    Bernardi, A., Brachat, J., Comon, P., Mourrain, B.: General tensor decomposition, moment matrices and applications. J. Symbolic Comput. 52, 51–71 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brachat, J., Comon, P., Mourrain, B., Tsigaridas, E.P.: Symmetric tensor decomposition. Linear Algebra Appl. 433(11–12), 1851–1872 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Comon, P., Jutten, C.: Handbook of Blind Source Separation: Independent Component Analysis and Applications. Academic press, Cambridge (2010)Google Scholar
  6. 6.
    Elkadi, M., Mourrain, B.: Introduction à la résolution des systèmes polynomiaux. Mathématiques et Applications, vol. 59. Springer, Heidelberg (2007)zbMATHGoogle Scholar
  7. 7.
    Harmouch, J., Khalil, H., Mourrain, B.: Structured low rank decomposition of multivariate Hankel matrices. Linear Algebra Appl. (2017).  https://doi.org/10.1016/j.laa.2017.04.015
  8. 8.
    Connelly, A., Tournier, J.D., Calamante, F.: Robust determination of the fiber orientation distribution in diffusion MRI: non-negativity constrained superresolved spherical deconvolution. NI 35(4), 1459–1472 (2007)Google Scholar
  9. 9.
    Jiang, T., Sidiropoulos, N.D.: Kruskal’s permutation lemma and the identification of candecomp/parafac and bilinear models with constant modulus constraints. IEEE Trans. Sig. Process. 52(9), 2625–2636 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Landsberg, J.M.: Tensors: Geometry and Applications. Graduate Studies in Mathematics. American Mathematical Society, Providence (2011)zbMATHGoogle Scholar
  11. 11.
    De Lathauwer, L., Castaing, J.: Tensor-based techniques for the blind separation of DS–CDMA signals. Sig. Process. 87(2), 322–336 (2007). Tensor Signal ProcessingGoogle Scholar
  12. 12.
    Megherbi, T., Kachouane, M., Boumghar, F.O., Deriche, R.: Détection des croisements de fibre en IRM de diffusion par décomposition de tenseur: Approche analytique. In: Reconnaissance de Formes et Intelligence Artificielle (RFIA), France, June 2014Google Scholar
  13. 13.
    Mourrain, B.: Isolated points, duality and residues. J. Pure Appl. Algebra 117&118, 469–493 (1996)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Mourrain, B.: Polynomial-exponential decomposition from moments. Found. Comput. Math. (2017).  https://doi.org/10.1007/s10208-017-9372-x
  15. 15.
    Roch, S.: A short proof that phylogenetic tree reconstruction by maximum likelihood is hard. IEEE/ACM Trans. Comput. Biol. Bioinform. 3(1), Page 92 (2006)Google Scholar
  16. 16.
    Sanchez, E., Kowalski, B.R.: Tensorial resolution: a direct trilinear decomposition. J. Chemometr. 4(1), 29–45 (1990)CrossRefGoogle Scholar
  17. 17.
    Sidiropoulos, N.D., Giannakis, G.B., Bro, R.: Blind parafac receivers for DS-CDMA systems. IEEE Trans. Sig. Process. 48(3), 810–823 (2000)CrossRefGoogle Scholar
  18. 18.
    Smilde, A., Bro, R., Geladi, P.: Multi-way Analysis: Applications in the Chemical Sciences. Wiley, Chichester (2005)Google Scholar
  19. 19.
    Atkins, M.S., Weldeselassie, T.Y., Barmpoutis, A.: Symmetric positive semi-definite Cartesian tensor fiber orientation distributions (CT-FOD). Med. Image Anal. J. 16(6), 1121–1129 (2012). Elsevier BVGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Laboratory of Mathematics and Its Applications LaMA-LebanonLebanese UniversityBeirutLebanon
  2. 2.UCA, Inria, AROMATHSophia AntipolisFrance

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