The Fr\'echet derivative of the tensor t-function

Abstract

The tensor t-function, a formalism that generalizes the well-known concept of matrix functions to third-order tensors, is introduced in [K. Lund, The tensor t-function: a definition for functions of third-order tensors, Numer. Linear Algebra Appl. 27 (3), e2288]. In this work, we investigate properties of the Fr\'echet derivative of the tensor t-function and derive algorithms for its efficient numerical computation. Applications in condition number estimation and nuclear norm minimization are explored. Numerical experiments implemented by the \texttt{t-Frechet} toolbox hosted at \url{https://gitlab.com/katlund/t-frechet} illustrate properties of the t-function Fr\'echet derivative, as well as the efficiency and accuracy of the proposed algorithms.