Let A be an ruth order n-dimensional tensor, where m, n are some positive integers and N ：= re（n-1）. Then A is called a Hankel tensor associated with a vector v ∈ R^N＋1 if Aσ = Vk for each k = 0,1,...,N whenever...Let A be an ruth order n-dimensional tensor, where m, n are some positive integers and N ：= re（n-1）. Then A is called a Hankel tensor associated with a vector v ∈ R^N＋1 if Aσ = Vk for each k = 0,1,...,N whenever σ= （i1,..., im） satisfies i1 ＋... ＋ im - m ＋ k. We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k = 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and （0,1） Hankel tensors. Keywords Tensor, convolution, Hankel tensor, elementary Hankel tensor, symmetric tensor展开更多
In this paper,an accelerated proximal gradient algorithm is proposed for Hankel tensor completion problems.In our method,the iterative completion tensors generated by the new algorithm keep Hankel structure based on p...In this paper,an accelerated proximal gradient algorithm is proposed for Hankel tensor completion problems.In our method,the iterative completion tensors generated by the new algorithm keep Hankel structure based on projection on the Hankel tensor set.Moreover,due to the special properties of Hankel structure,using the fast singular value thresholding operator of the mode-s unfolding of a Hankel tensor can decrease the computational cost.Meanwhile,the convergence of the new algorithm is discussed under some reasonable conditions.Finally,the numerical experiments show the effectiveness of the proposed algorithm.展开更多
We extend Vandermonde matrices to generalized Vandermonde tensors. We call an ruth order n-dimensional real tensor A = （Ai1i2…im） a type-1 generalized Vandermonde （GV） tensor, or GV1 tensor, if there exists a vec...We extend Vandermonde matrices to generalized Vandermonde tensors. We call an ruth order n-dimensional real tensor A = （Ai1i2…im） a type-1 generalized Vandermonde （GV） tensor, or GV1 tensor, if there exists a vector v = （v1, v2,.. , Vn）T such that Aili2...im = vi2＋i3＋＋im-m＋l and call A , a type-2 （ruth order n dimensional） GV tensor, or GV2 tensor, if there exists an （m - 1）th order tensor B= （Bi1i2…im-1） such that Ai1i2…im= Bim-1i1i2…im In this paper, we mainly investigate the type-1 GV tensors including their products, their spectra, and their positivities. Applications of GV tensors are also introduced.展开更多
文摘Let A be an ruth order n-dimensional tensor, where m, n are some positive integers and N ：= re（n-1）. Then A is called a Hankel tensor associated with a vector v ∈ R^N＋1 if Aσ = Vk for each k = 0,1,...,N whenever σ= （i1,..., im） satisfies i1 ＋... ＋ im - m ＋ k. We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k = 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and （0,1） Hankel tensors. Keywords Tensor, convolution, Hankel tensor, elementary Hankel tensor, symmetric tensor
文摘In this paper,an accelerated proximal gradient algorithm is proposed for Hankel tensor completion problems.In our method,the iterative completion tensors generated by the new algorithm keep Hankel structure based on projection on the Hankel tensor set.Moreover,due to the special properties of Hankel structure,using the fast singular value thresholding operator of the mode-s unfolding of a Hankel tensor can decrease the computational cost.Meanwhile,the convergence of the new algorithm is discussed under some reasonable conditions.Finally,the numerical experiments show the effectiveness of the proposed algorithm.
文摘We extend Vandermonde matrices to generalized Vandermonde tensors. We call an ruth order n-dimensional real tensor A = （Ai1i2…im） a type-1 generalized Vandermonde （GV） tensor, or GV1 tensor, if there exists a vector v = （v1, v2,.. , Vn）T such that Aili2...im = vi2＋i3＋＋im-m＋l and call A , a type-2 （ruth order n dimensional） GV tensor, or GV2 tensor, if there exists an （m - 1）th order tensor B= （Bi1i2…im-1） such that Ai1i2…im= Bim-1i1i2…im In this paper, we mainly investigate the type-1 GV tensors including their products, their spectra, and their positivities. Applications of GV tensors are also introduced.
