Generalized tensor function via the tensor singular value decomposition based on the T-product

By Yun Miao, Liqun Qi, Yimin Wei. Linear Algebra and its Applications, 2019

145 citations recorded by the source corpus.

In this paper, we present the definition of generalized tensor function according to the tensor singular value decomposition (T-SVD) via the tensor T-product. Also, we introduce the compact singular value decomposition (T-CSVD) of tensors via the T-product, from which the projection operators and Moore Penrose inverse of tensors are also obtained. We also establish the Cauchy integral formula for tensors by using the partial isometry tensors and applied it into the solution of tensor equations. Then we establish the generalized tensor power and the Taylor expansion of tensors. Explicit generalized tensor functions are also listed. We define the tensor bilinear and sesquilinear forms and proposed 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 established 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 also raised.

Subfields: Artificial Intelligence, Computational Mathematics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/f4d758ba66a16ee84f93006cd621148c262a1714

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Yimin Wei
2. Yun Miao
3. Liqun Qi

In Computational Mathematics

1. Yimin Wei
2. Liqun Qi
3. Yun Miao
4. Chaoqian Li
5. Yaotang Li
6. Yisheng Song
7. J. Shao
8. Guanglu Zhou

In Signal Processing

1. Yimin Wei
2. Yun Miao
3. Liqun Qi

Citations in the corpus

• T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product (2019, methodology): “The concept of T-range space, T-null space, tensor norm, and T-Moore-Penrose inverse is given as follows [50].”