379 research outputs found
SOS-Hankel Tensors: Theory and Application
Hankel tensors arise from signal processing and some other applications. SOS
(sum-of-squares) tensors are positive semi-definite symmetric tensors, but not
vice versa. The problem for determining an even order symmetric tensor is an
SOS tensor or not is equivalent to solving a semi-infinite linear programming
problem, which can be done in polynomial time. On the other hand, the problem
for determining an even order symmetric tensor is positive semi-definite or not
is NP-hard. In this paper, we study SOS-Hankel tensors. Currently, there are
two known positive semi-definite Hankel tensor classes: even order complete
Hankel tensors and even order strong Hankel tensors. We show complete Hankel
tensors are strong Hankel tensors, and even order strong Hankel tensors are
SOS-Hankel tensors. We give several examples of positive semi-definite Hankel
tensors, which are not strong Hankel tensors. However, all of them are still
SOS-Hankel tensors. Does there exist a positive semi-definite non-SOS-Hankel
tensor? The answer to this question remains open. If the answer to this
question is no, then the problem for determining an even order Hankel tensor is
positive semi-definite or not is solvable in polynomial-time. An application of
SOS-Hankel tensors to the positive semi-definite tensor completion problem is
discussed. We present an ADMM algorithm for solving this problem. Some
preliminary numerical results on this algorithm are reported
Four lectures on secant varieties
This paper is based on the first author's lectures at the 2012 University of
Regina Workshop "Connections Between Algebra and Geometry". Its aim is to
provide an introduction to the theory of higher secant varieties and their
applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in
Mathematics & Statistics), Springer/Birkhause
A note on the gap between rank and border rank
We study the tensor rank of the tensor corresponding to the algebra of
n-variate complex polynomials modulo the dth power of each variable. As a
result we find a sequence of tensors with a large gap between rank and border
rank, and thus a counterexample to a conjecture of Rhodes. At the same time we
obtain a new lower bound on the tensor rank of tensor powers of the generalised
W-state tensor. In addition, we exactly determine the tensor rank of the tensor
cube of the three-party W-state tensor, thus answering a question of Chen et
al.Comment: To appear in Linear Algebra and its Application
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