Abstract
The notation of Moore-Penrose inverse of matrices has been extended from matrix space to even-order tensor space with Einstein product. In this paper, we give the numerical study on the Moore-Penrose inverse of tensors via the Einstein product. More precisely, we transform the calculation of Moore-Penrose inverse of tensors via the Einstein product into solving a class of tensor equations via the Einstein product. Then, by means of the conjugate gradient method, we obtain the approximate Moore-Penrose inverse of tensors via the Einstein product. Finally, we report some numerical examples to show the efficiency of the proposed methods and testify the conclusion suggested in this paper.
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Bader, B.W., Kolda, T.G., et al.: MATLAB Tensor Toolbox Version 2.5, 2012. http://www.sandia.gov/tgkolda/TensorToolbox/
Behera, R., Maji, S., Mohapatra, R.N.: Weighted Moore-Penrose inverses of arbitrary order tensors. Comput. Appl. Math. 39, 284 (2020)
Behera, R., Mishra, D.: Further results on generalized inverses of tensors via the Einstein product. Linear Multilinear Algebra 65, 1662–1682 (2017)
Brazell, M., Li, N., Navasca, C., Tamon, C.: Solving multilinear systems via tensor inversion. SIAM J. Matrix Anal. Appl. 34, 542–570 (2013)
Bu, C., Zhang, X., Zhou, J., Wang, W., Wei, Y.: The inverse, rank and product of tensors. Linear Algebra Appl. 446, 269–280 (2014)
Bu, C., Zhou, J., Wei, Y.: E-cospectral hypergraphs and some hypergraphs determined by their spectra. Linear Algebra Appl. 459, 397–403 (2014)
Burdick, D.S., Tu, X.M., McGown, L.B., Millican, D.W.: Resolution of multicomponent fluorescent mixtures by analysis of the excitation-emission frequency array. J. Chemom. 4, 15–28 (1990)
Cooper, J., Dutle, A.: Spectra of uniform hypergraphs. Linear Algebra Appl. 436, 3268–3292 (2012)
Ding, W., Wei, Y.: Fast Hankel tensor-vector product and its application to exponential data fitting. Numer. Linear Algebra Appl. 22, 814–832 (2015)
Eldén, L.: Matrix Methods in Data Mining and Pattern Recognition. SIAM, Philadelphia (2007)
Hu, S., Qi, L.: Algebraic connectivity of an even uniform hypergraph. J. Comb. Optim. 24, 564–579 (2012)
Huang, B.H., Ma, C.F.: An iterative algorithm to solve the generalized Sylvester tensor equations. Linear Multilinear Algebra 68, 1175–1200 (2020)
Huang, B.H., Xie, Y.J., Ma, C.F.: Krylov subspace methods to solve a class of tensor equations via the Einstein product. Numer. Linear Algebra Appl. 26, e2254 (2019)
Huang, B.H., Ma, C.F.: Global least squares methods based on tensor form to solve a class of generalized Sylvester tensor equations. Appl. Math. Comput. 369, 124892 (2020)
Huang, B.H., Li, W.: Numerical subspace algorithms for solving the tensor equations involving Einstein product. Numer. Linear Algebra Appl (2020). https://doi.org/10.1002/nla.2351
Kolda, T., Bader, B.: Tensor decompositions and applications. SIAM Rev. 51, 455–500 (2009)
Lai, W.M., Rubin, D., Krempl, E.: Introduction to Continuum Mechanics. Butterworth Heinemann, Oxford (2009)
Li, B.W., Tian, S., Sun, Y.S., Hu, Z.M.: Schur-decomposition for 3D matrix equations and its applications in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method. J. Comput. Phys. 229, 1198–1212 (2010)
Li, B.W., Sun, Y.S., Zhang, D.W.: Chebyshev collocation spectral methods for coupled radiation and conduction in a concentric spherical participating medium. ASME J. Heat Transfer. 131, 062701–062709 (2009)
Li, W., Ng, M.: On the limiting probability distribution of a transition probability tensor. Linear Multilinear Algebra 62, 362–385 (2014)
Li, Z., Ling, C., Wang, Y., Yang, Q.: Some advances in tensor analysis and polynomial optimization. Oper. Res. Trans. 18, 134–148 (2014)
Ji, J., Wei, Y.: Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product. Front. Math. China 12, 1319–1337 (2017)
Ji, J., Wei, Y.: The Drazin inverse of an even-order tensor and its application to singular tensor equations. Comput. Math. Appl. 75, 3402–3413 (2018)
Jin, H., Bai, M., Bentez, J., Liu, X.: The generalized inverses of tensors and an application to linear models. Comput. Math. Appl. 74, 385–397 (2017)
Liang, M., Zheng, B.: Further results on Moore-Penrose inverses of tensors with application to tensor nearness problems. Comput. Math. Appl. 77, 1282–1293 (2019)
Ma, H.F., Li, N., Stanimirović, P.S., Katsikis, V.N.: Perturbation theory for Moore-Penrose inverse of tensor via Einstein product. Comput. Appl. Math. 38, 111 (2019)
Martin, C.D., Shafer, R., LaRue, B.: An order-p tensor factorization with applications in imaging. SIAM J. Sci. Comput. 35, 474–490 (2013)
Panigrahy, K., Behera, R., Mishra, D.: Reverse-order law for the Moore-Penrose inverses of tensors. Linear Multilinear Algebra 68, 246–264 (2020)
Panigrahy, K., Mishra, D.: On reverse-order law of tensors and its application to additive results on Moore–Penrose inverse. RACSAM 184, 114 (2020)
Panigrahy, K., Mishra, D.: Extension of Moore-Penrose inverse of tensor via Einstein product. Linear Multilinear Algebra (2020) https://doi.org/10.1080/03081087.2020.1748848
Smilde, A., Bro, R., Geladi, P.: Multi-Way Analysis: Applications in the Chemical Sciences. Wiley, West Sussex (2004)
Sun, L., Zheng, B., Bu, C., Wei, Y.: Moore-Penrose inverse of tensors via Einstein product. Linear Multilinear Algebra 64, 686–698 (2016)
Vlasic, D., Brand, M., Pfister, H., Popovic, J.: Face transfer with multilinear models. ACM Trans. Graph. 24, 426–433 (2005)
Wang, Q.W., Xu, X.: Iterative algorithms for solving some tensor equations. Linear Multilinear Algebra 67, 1325–1349 (2019)
Funding
This research is supported by China Postdoctoral Science Foundation (Grant No. 2019M660203) and National Natural Science Foundation of China (Grant Nos. 12001211, 12071159, 61976053)
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Huang, B. Numerical study on Moore-Penrose inverse of tensors via Einstein product. Numer Algor 87, 1767–1797 (2021). https://doi.org/10.1007/s11075-021-01074-0
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DOI: https://doi.org/10.1007/s11075-021-01074-0