The tensor t‐function: A definition for functions of third‐order tensors

Abstract

A definition for functions of multidimensional arrays is presented. The definition is valid for third-order tensors in the tensor t-product formalism, which regards third-order tensors as block circulant matrices. The tensor function definition is shown to have similar properties as standard matrix function definitions in fundamental scenarios. To demonstrate the definition's potential in applications, the notion of network communicability is generalized to third-order tensors and computed for a small-scale example via block Krylov subspace methods for matrix functions. A complexity analysis for these methods in the context of tensors is also provided. K E Y W O R D Sblock circulant matrices, matrix functions, multidimensional arrays, network analysis, tensor t-product, tensors

T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

“…By using the concept of T-product, the matrix function can be generalized to tensors of size n × n × p. Suppose we have tensors A ∈ C n×n×p and B ∈ C n×s×p , then the tensor T-function of A is defined by [26]:…”

“…As an direct application of T-Jordan canonical form, we discuss the F-square tensor power series as an extension of Lund's results [26] and Theorem 2.…”

“…An approach of linearization is provided by the T-product to transfer tensor multiplication to matrix multiplication by the discrete Fourier transformation and the theories of block circulant matrices [6,19]. Due to the importance of the tensor T-product, Lund [26] gave the definition of tensor functions based on the T-product of third-order F-square tensors in her Ph.D thesis in 2018. The definition of T-function is given by…”

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

“…Recently, the tensor T-product has been established and proved to be a useful tool in many areas, such as image processing [28,29,42,47,50,60], computer vision [4,17,53,55], signal processing [10,34,37,48], low rank tensor recovery and robust tensor PCA [32,34], data completion and denoising [12,25,26,35,37,39,41,45,51,54,56,57,58,59]. Because of the importance of tensor T-product, Lund [38] gave the definition for tensor functions based on the T-product of third-order F-square tensors which means all the front slices of a tensor is square matrices. The definition of T-function is given by where 'bcirc(A)' is the block circulant matrix [9] by the F-square tensor A ∈ R n×n×p and see the detail in Section 2.2.…”

“…We make some review of the definition of tensor T-product and some algebraic structure of third order tensors via this kind of product in Preliminaries. Then we recall the definition of T-function given by Lund [38] and some of its properties. In the main part of our paper, we introduce the definition of tensor Figure 1: T-SVD of Tensors singular value decomposition and the generalized matrix functions.…”