In this paper, we present the definition of generalized tensor function according to the tensor singular value decomposition (T-SVD) based on the tensor T-product. Also, we introduce the compact singular value decompo...
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In this paper, we present the definition of generalized tensor function according to the tensor singular value decomposition (T-SVD) based on the tensor T-product. Also, we introduce the compact singular value decomposition (T-CSVD) of tensors, from which the projection operators and Moore-Penrose inverse of tensors are obtained. We establish the Cauchy integral formula for tensors by using the partial isometry tensors and apply it into the solution of tensor equations. Then we establish the generalizedtensor power and the Taylor expansion of tensors. Explicit generalized tensor functions are listed. We define the tensor bilinear and sesquilinear forms and propose theorems on structures preserved by generalized tensor functions. For complex tensors, we established an isomorphism between complex tensors and real tensors. In the last part of our paper, we find that the block circulant operator establishes an isomorphism between tensors and matrices. This isomorphism is used to prove the F-stochastic structure is invariant under generalized tensor functions. The concept of invariant tensor cones is raised. (C) 2020 Elsevier Inc. All rights reserved.
A large number of real-world problems can be transformed into mathematical problems by means of third-order real tensors. Recently, as an extension of the generalized matrix function, the generalized tensor function o...
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A large number of real-world problems can be transformed into mathematical problems by means of third-order real tensors. Recently, as an extension of the generalized matrix function, the generalized tensor function over the third-order real tensor space was introduced with the aid of a scalar function based on the T-pro duct for third-order tensors and the tensor singular value decomposition;and some useful algebraic properties of the function were investigated. In this paper, we show that the generalized tensor function can inherit a lot of good properties from the associated scalar function, including continuity, directional differentiability, Fre ' chet differentiability, Lipschitz continuity and semismoothness. These properties provide an important theoretical basis for the studies of various mathematical problems with generalized tensor functions, and particularly, for the studies of tensor optimization problems with generalized tensor functions.
This paper is concerned with Krylov subspace methods based on the tensor t-product for computing certain quantities associated with generalized third-order tensorfunctions. We use the tensor t-product and define the ...
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This paper is concerned with Krylov subspace methods based on the tensor t-product for computing certain quantities associated with generalized third-order tensorfunctions. We use the tensor t-product and define the tensor global Golub-Kahan bidiagonalization process for approximating tensorfunctions. Pairs of Gauss and Gauss-Radau quadrature rules are applied to determine the desired quantities with error bounds. An application to the computation of the tensor nuclear norm is presented and illustrates the effectiveness of the proposed methods.
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