Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares (SOS) algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors (EEN-tensors), which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a nonnegative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.
BarmpoutisA, VemuriB C, HowlandD, ForderJ R. Regularized positive-definite fourth order tensor field estimation from DW-MRI.Neuroimage, 2009, 45: 153–162 https://doi.org/10.1016/j.neuroimage.2008.10.056
2
BasserP J, JonesD K. Diffusion-tensor MRI: theory, experimental design and data analysis a technical review.NMR Biomed, 2002, 15: 456–467 https://doi.org/10.1002/nbm.783
DingW, QiL, WeiY. Fast Hankel tensor-vector product and its application to exponential data fitting.Numer Linear Algebra Appl,2015, 22: 814–832 https://doi.org/10.1002/nla.1970
11
HuS, LiG, QiL. A tensor analogy of Yuan’s alternative theorem and polynomial optimization with sign structure.J Optim Theory Appl, 2016, 168: 446–474 https://doi.org/10.1007/s10957-014-0652-1
12
HuS, LiG, QiL, SongY. Finding the maximum eigenvalue of essentially nonnegative symmetric tensors via sum of squares programm ing.J Optim Theory Appl,2013, 158: 717–738 https://doi.org/10.1007/s10957-013-0293-9
13
HuS, QiL, XieJ. The largest Laplacian and signless Laplacian H-eigenvalues of a uniform hypergraph.Linear Algebra Appl,2015, 469: 1–27 https://doi.org/10.1016/j.laa.2014.11.020
14
LiuY, ZhouG, IbrahimN F. An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor.J Comput Appl Math, 2010, 235(1): 286–292 https://doi.org/10.1016/j.cam.2010.06.002
15
LöfbergJ. YALMIP: a toolbox for modeling and optimization in MATLAB.In: IEEE International Symposium on Computer Aided Control Systems Design, Taipei, September24, 2004. 2004, 284–289
16
LöfbergJ. Pre- and post-processing sums-of-squares programs in practice.IEEE Trans Automat Control, 2009, 54: 1007–1011 https://doi.org/10.1109/TAC.2009.2017144
17
NgM, QiL, ZhouG. Finding the largest eigenvalue of a non-negative tensor.SIAM J Matrix Anal Appl, 2009, 31: 1090–1099 https://doi.org/10.1137/09074838X
WangY, CaccettaL, ZhouG. Convergence analysis of a block improvement method for polynomial optimization over unit spheres.Numer Linear Algebra Appl,2015, 22: 1059–1076 https://doi.org/10.1002/nla.1996
24
WangY, LiuW, CaccettaL, ZhouG. Parameter selection for nonnegative l1 matrix/tensor sparse decomposition.Oper Res Lett, 2015, 43: 423–426 https://doi.org/10.1016/j.orl.2015.06.005
25
WangY, QiL, ZhangX. A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor.Numer Linear Algebra Appl,2009, 16: 589–601 https://doi.org/10.1002/nla.633
ZhangK, WangY. An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms.J Comput Appl Math, 2016, 305: 1–10 https://doi.org/10.1016/j.cam.2016.03.025
