On spectral data and tensor decompositions in Finslerian framework

Document Type : Original Article

Author

Department Mathematics-Informatics, Faculty of Applied Sciences, University Politehnica of Bucharest

Abstract

The extensions of the Riemannian structure include the Finslerian one, which provided in recent years successful models in various fields like Biology, Physics, GTR, Monolayer Nanotechnology and Geometry of Big Data. The present article provides the necessary notions on tensor spectral data and on the HO-SVD and the Candecomp tensor decompositions, and further study several aspects related to the spectral theory of the main symmetric Finsler tensors, the fundamental and the Cartan tensor. In particular, are addressed two Finsler models used in Langmuir Blodgett Nanotechnology and in Oncology. As well, the HO-SVD and Candecomp decompositions are exemplified for these models and metric extensions of the eigen problem are proposed.

Keywords

Main Subjects


[1] K. C. Chang, K. Pearson and T. Zhang, On eigenvalue problems of real symmetric tensors, Journal of Mathematical Analysis and Applications 350 (2009), 416-422.
[2] L. Qi, Eigenvalues of a real supersymmetric tensor, Jour. Symb. Comp. 40 (2005), 1302-1324.
[3] A. Cichocki, N. Lee, I. V. Oseledets, A. H. Phan, Q. Zhao and D. P. Mandic, Tensor networks for dimensionality reduction and large-scale optimization: part 1 low-rank tensor decompositions, Found. Trends Mach. Learn. 9 (4-5) (2016), 249-429.
[4] P. Comon, Tensor diagonalization, a useful tool in signal processing, In: IFAC-SYSID, 10th IFAC Symposium on System Identification (Copenhagen, Denmark, July 4-6, 1994. Invited Session), Blanke M., Soderstrom T. (eds), 1 (1994), 77-82.
[5] B. Jiang, S. Ma and S. Zhang, Tensor principal component analysis via convex optimization, Mathematical Programming 150, 2 (2015), 423-457.
[6] L. de Lathauwer, B. de Moor and J. Vandewalle, A multilinear SVD, SIAM J. Matrix Anal. Appl. 21 (2000), 1253-1278.
[7] L. Qi, W. Sun and Y. Wang, Numerical multilinear algebra and its applications, Front. Math. China 2(4) (2007), 501-526.
[8] N. Vannieuwenhoven, N. Vanbaelen, K. Meerbergen and R. Vandebril, The dense multiple-vector tensor-vector product: An initial study, Report TW 635, Katholieke Universiteit Leuven, Department of Computer Science, 2013.
[9] Z. Shen, Differential geometry of spray and Finsler spaces, Springer, 2001.
[10] D. Bao, S-S. Chern and Z. Shen, An introduction to Riemann-Finsler geometry, Springer-Verlag, 2000.
[11] I. Bucataru and R. Miron, Finsler-Lagrange geometry. Applications to dynamical systems, Editura Academiei Romane, Bucuresti, 2007.
[12] X. Cheng and Z. Shen, Finsler Geometry: An Approach via Randers Spaces, Springer, 2013.
[13] R. Miron and M. Anastasiei, Vector bundles. Lagrange spaces. Applications to relativity, Geometry Balkan Press, 1996.
[14] V. Balan, Spectra of multilinear forms associated to notable m-root relativistic models, Linear Algebra and Appl. (LAA), online http://dx.doi.org/10.1016/j.laa.2011.06.033; 436, 1, 1 (2012), 152-162. [15] V. Balan, Spectral properties and applications of numerical multilinear algebra of m−th root structures, Hypercomplex Numbers in Geom. Phys. 2(10), 5 (2008), 101-107.
[16] V. Balan, G. Bogoslovsky, S. Kokarev, D. Pavlov, S. Syparov and N. Voicu, Geometrical models of the locally anisotropic Space-Time, Hypercomplex Numbers in Geom. Phys., Moscow, 1(15), 8 (2011), 4-37.
[17] M. D. Cirillo, R. Mirdell, F. Sjoberg and T. Pham, Tensor decomposition for colour image segmentation of burn wounds, Sci. Rep. 9 (2019), 3291.
[18] Y. Zhang, J. Liu, S. Yang and Z. Guo, Joint image denoising using self-similarity based low-rank approxima[1]tions, Institute of Computer Science and Technology, Peking University, Beijing 100871, China; 2013 Visual Communications and Image Processing (VCIP).
[19] D. Goldfarb and Z. (Tony) Qin, Robust Low-Rank Tensor Recovery: Models and Algorithms, SIAM J. Matrix Anal. Appl., 35(1) (2019), 225-253.
[20] S. Zhang, X. Guo, X. Xu, L. Li andC-C Chang, A video watermark algorithm based on tensor decomposition, Math Biosci Eng. 16(5) (2019), 3435-3449.
[21] J. Li, J. Bien and M. Wells, HO-SVD package, 2019; https://rdrr.io/github/jamesyili/rTensor/
[22] , MATLAB Tensor Toolbox, Version 2.6.
[23] , Tensorlab demos, www.tensorlab.net/demos/basic.html, www.tensorlab.net/demos/mlsvd.html.
[24] , Sandia Laboratories webpage, https://www.sandia.gov/ tgkolda/TensorToolbox/reg-2.6.html
[25] , Tensor Toolbox webpage, https://www.tensortoolbox.org
[26] , Htucker webpage (Matlab toolbox for HO-SVD), Czech Republic, https://www.swmath.org/software/9637
[27] V. Balan and N. Perminov, Applications of resultants in the spectral m-root framework, Appl. Sci. (APPS), 12 (2010), 20-29.
[28] V. Balan, H. V. Grushevskaya, N. G. Krylova, M. Neagu and A. Oana, On the Berwald-Lagrange scalar curvature in the structuring process of the LB-monolayer, Applied Sciences 15 (2013), 30-42.
[29] V. Balan, H. V. Grushevskaya and N. G. Krylova, Finsler geometry approach to thermodynamics of first order phase transitions in monolayers, Differential Geometry - Dynamical Systems, 17 (2015), 24-31.
[30] V. Balan and J. Stojanov, Finsler-type estimators for the cancer cell population dynamics, Publications de l’Institut Mathematique, Publisher: Mathematical Institute of the Serbian Academy of Sciences and Arts, Beograd, 98(112) (2015), 53-69.
[31] V. Balan, J. Stojanov, Anisotropic metric models in the Garner oncologic framework, ROMAI J. 10, 1 (2014), 1-10.
[32] L. Astola and L. Florack, Finsler Geometry on higher order tensor fields and applications to High Angular Res[1]olution Diffusion Imaging, International Journal of Computer Vision 5567 (3) (2009), 224-234, DOI:10.1007/978- 3-642-02256-2 19.
[33] H. Shimada, On Finsler spaces with the metric L = pn ai1i2...in (x)y i1 y i2 . . . yin , Tensor N.S., 33 (1979), 365-372.
[34] P. L. Antonelli, R. S. Ingarden and M. Matsumoto, The theory of sprays and Finsler spaces with applications in physics and biology, Kluwer Academic Publishers, Fundamental Theories of Physics 58, Netherlands, 1993.