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 -