An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms

Abstract Identifying the positive definiteness of an even-order homogeneous multivariate form is an important task due to its wide applications in such as medical imaging and the stability analysis of nonlinear autonomous systems via Lyapunov’s direct method in automatic control and multivariate network realizability analysis. In this paper, based on the equivalence of the positive definiteness of the form to that of the underlying tensor, and the links between the positive definiteness of a tensor with strong H -tensor, we propose an H -tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms. The validity of the iterative scheme is guaranteed theoretically and the given numerical experiments show the efficiency of the scheme.

[1]  L. Qi,et al.  Higher Order Positive Semidefinite Diffusion Tensor Imaging , 2010, SIAM J. Imaging Sci..

[2]  Kung-Ching Chang,et al.  Perron-Frobenius theorem for nonnegative tensors , 2008 .

[3]  L. Qi,et al.  Strictly nonnegative tensors and nonnegative tensor partition , 2011, Science China Mathematics.

[4]  Derek K. Jones,et al.  Diffusion‐tensor MRI: theory, experimental design and data analysis – a technical review , 2002 .

[5]  M. A. Hasan,et al.  A procedure for the positive definiteness of forms of even order , 1996, IEEE Trans. Autom. Control..

[6]  Yiju Wang,et al.  Criteria for strong H-tensors , 2016 .

[7]  Qingzhi Yang,et al.  Further Results for Perron-Frobenius Theorem for Nonnegative Tensors , 2010, SIAM J. Matrix Anal. Appl..

[8]  L. Qi,et al.  Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors , 2014, 1405.6363.

[9]  A. Berman,et al.  Some properties of strong H-tensors and general H-tensors , 2015 .

[10]  Fei Wang,et al.  Comments on "Explicit criterion for the positive definiteness of a general quartic form" , 2005, IEEE Trans. Autom. Control..

[11]  S. Gaubert,et al.  Perron–Frobenius theorem for nonnegative multilinear forms and extensions , 2009, 0905.1626.

[12]  Yan Zhu,et al.  Criterions for the positive definiteness of real supersymmetric tensors , 2014, J. Comput. Appl. Math..

[13]  N. Bose,et al.  General procedure for multivariable polynomial positivity test with control applications , 1976 .

[14]  Liqun Qi,et al.  M-Tensors and Some Applications , 2014, SIAM J. Matrix Anal. Appl..

[15]  Baba C. Vemuri,et al.  Regularized positive-definite fourth order tensor field estimation from DW-MRI , 2009, NeuroImage.

[16]  Yao-tang Li,et al.  On the iterative criterion for strong $$\mathcal {H}$$H-tensors , 2017 .

[17]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[18]  P. Basser,et al.  Estimation of the effective self-diffusion tensor from the NMR spin echo. , 1994, Journal of magnetic resonance. Series B.

[19]  Liqun Qi,et al.  Necessary and sufficient conditions for copositive tensors , 2013, 1302.6084.

[20]  L. Qi,et al.  Finding the Spectral Radius of a Nonnegative Tensor , 2011, 1111.2138.

[21]  Maher Moakher.  On the Averaging of Symmetric Positive-Definite Tensors , 2006 .

[22]  Andrzej Cichocki,et al.  Nonnegative Matrix and Tensor Factorization T , 2007 .

[23]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[24]  Liqun Qi,et al.  D-eigenvalues of diffusion kurtosis tensors , 2008 .

[25]  Yiju Wang,et al.  Nonsingular $H$-tensor and its criteria , 2016 .

[26]  L. Qi,et al.  M-tensors and nonsingular M-tensors , 2013, 1307.7333.

[27]  M.A. Hulsen.  A sufficient condition for a positive definite configuration tensor in differential models , 1990 .

[28]  Fei Wang,et al.  An Eigenvalue Method for Testing Positive Definiteness of a Multivariate Form , 2008, IEEE Transactions on Automatic Control.