Communications in Mathematical Sciences

Volume 14 (2016)

Number 8

SOS tensor decomposition: Theory and applications

Pages: 2073 – 2100



Haibin Chen (Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong)

Guoyin Li (Department of Applied Mathematics, University of New South Wales, Sydney, Australia)

Liqun Qi (Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong)


In this paper, we examine structured tensors which have sum-of-squares (SOS) tensor decomposition, and study the SOS-rank of SOS tensor decomposition. We first show that several classes of even order symmetric structured tensors available in the literature have SOS tensor decomposition. These include positive Cauchy tensors, weakly diagonally dominated tensors, $B_0$-tensors, double $B$-tensors, quasi-double $B_0$-tensors, $MB_0$-tensors, $H$-tensors, absolute tensors of positive semi-definite $Z$-tensors, and extended $Z$-tensors. We also examine the SOS-rank of SOS tensor decompositions and the SOS-width for SOS tensor cones. The SOS-rank provides the minimal number of squares in the SOS tensor decomposition, and, for a given SOS tensor cone, its SOS-width is the maximum possible SOS-rank for all the tensors in this cone. We first deduce an upper bound for general tensors that have SOS decomposition and the SOS-width for general SOS tensor cone using the known results in the literature of polynomial theory. Then, we provide an explicit sharper estimate for the SOS-rank of SOS tensor decomposition with bounded exponent and identify the SOS-width for the tensor cone consisting of all tensors with bounded exponent that have SOS decompositions. Finally, as applications, we show how the SOS tensor decomposition can be used to compute the minimum $H$-eigenvalue of an even order symmetric extended $Z$-tensor and test the positive definiteness of an associated multivariate form. Numerical examples ranging from small size to large size are provided to show the efficiency of the proposed numerical methods.


structured tensor, SOS tensor decomposition, positive semi-definite tensor, SOS-rank, $H$-eigenvalue

2010 Mathematics Subject Classification

15A06, 90C30

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