The tensor t-function: a definition for functions of third-order tensors

TL;DR

This paper introduces a new way to define functions of third-order tensors using the tensor t-product formalism, extending matrix function concepts to multidimensional arrays with potential applications in network analysis.

Contribution

It proposes a novel tensor function definition based on the tensor t-product, aligning tensor functions with matrix functions and demonstrating its application to network communicability.

Findings

Tensor function definition generalizes matrix functions to third-order tensors.

The approach enables computation of tensor functions via block Krylov subspace methods.

Complexity analysis supports practical application of the proposed methods.

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.