TY - JOUR
T1 - Inheritance properties and sum-of-squares decomposition of Hankel tensors
T2 - theory and algorithms
AU - Ding, Weiyang
AU - Qi, Liqun
AU - Wei, Yimin
N1 - Funding Information:
The first and the third authors are supported by the National Natural Science Foundation of China under Grant 11271084. The second author is supported by the Hong Kong Research Grant Council (Grant No. PolyU 502111, 501212, 501913 and 15302114).
PY - 2017/3/1
Y1 - 2017/3/1
N2 - In this paper, we show that if a lower-order Hankel tensor is positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS), then its associated higher-order Hankel tensor with the same generating vector, where the higher order is a multiple of the lower order, is also positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS, respectively). Furthermore, in this case, the extremal H-eigenvalues of the higher order tensor are bounded by the extremal H-eigenvalues of the lower order tensor, multiplied with some constants. Based on this inheritance property, we give a concrete sum-of-squares decomposition for each strong Hankel tensor. Then we prove the second inheritance property of Hankel tensors, i.e., a Hankel tensor has no negative (or non-positive, or positive, or nonnegative) H-eigenvalues if the associated Hankel matrix of that Hankel tensor has no negative (or non-positive, or positive, or nonnegative, respectively) eigenvalues. In this case, the extremal H-eigenvalues of the Hankel tensor are also bounded by the extremal eigenvalues of the associated Hankel matrix, multiplied with some constants. The third inheritance property of Hankel tensors is raised as a conjecture.
AB - In this paper, we show that if a lower-order Hankel tensor is positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS), then its associated higher-order Hankel tensor with the same generating vector, where the higher order is a multiple of the lower order, is also positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS, respectively). Furthermore, in this case, the extremal H-eigenvalues of the higher order tensor are bounded by the extremal H-eigenvalues of the lower order tensor, multiplied with some constants. Based on this inheritance property, we give a concrete sum-of-squares decomposition for each strong Hankel tensor. Then we prove the second inheritance property of Hankel tensors, i.e., a Hankel tensor has no negative (or non-positive, or positive, or nonnegative) H-eigenvalues if the associated Hankel matrix of that Hankel tensor has no negative (or non-positive, or positive, or nonnegative, respectively) eigenvalues. In this case, the extremal H-eigenvalues of the Hankel tensor are also bounded by the extremal eigenvalues of the associated Hankel matrix, multiplied with some constants. The third inheritance property of Hankel tensors is raised as a conjecture.
KW - Convolution
KW - Hankel tensor
KW - Inheritance property
KW - Positive semi-definite tensor
KW - Sum-of-squares
UR - http://www.scopus.com/inward/record.url?scp=85012914511&partnerID=8YFLogxK
U2 - 10.1007/s10543-016-0622-0
DO - 10.1007/s10543-016-0622-0
M3 - Journal article
AN - SCOPUS:85012914511
SN - 0006-3835
VL - 57
SP - 169
EP - 190
JO - BIT Numerical Mathematics
JF - BIT Numerical Mathematics
IS - 1
ER -