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

TL;DR

This paper studies the Fréchet derivative of the tensor t-function, extending matrix function concepts to third-order tensors, and develops algorithms for its efficient computation with applications in condition estimation and norm minimization.

Contribution

It introduces properties of the Fréchet derivative of the tensor t-function and provides algorithms for its efficient numerical computation, with practical applications and numerical validation.

Findings

Algorithms demonstrate high efficiency and accuracy.

Applications include condition number estimation and nuclear norm minimization.

Numerical experiments validate theoretical properties and computational methods.

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.