Conjugate gradient-like methods for solving general tensor equation with Einstein product

Abstract

Tensors have a wide application in data mining, chemistry, information sciences, documents analysis and medical engineering. In this work, we study the general tensor equation ∑i=1lFi*PX*QGi=H with Einstein product where Fi,Gi,H, for i=1,2,…,l, are known tensors and X is an unknown tensor to be determined. The main motivation for this study is the investigation of conjugate gradient-like methods for solving this tensor equation. We show that the conjugate gradient-like methods converge to tensor solutions in a finite number of steps in the absence of round-off errors. Numerical examples confirm ’oretical results and demonstrate the accuracy and computational efficiency of the methods.