Orthogonal Tensor Decompositions

Abstract

We explore the orthogonal decomposition of tensors (also known as multidimensional arrays or n-way arrays) using two different definitions of orthogonality. We present numerous examples to illustrate the difficulties in understanding such decompositions. We conclude with a counterexample to a tensor extension of the Eckart–Young SVD approximation theorem by Leibovici and Sabatier [Linear Algebra Appl., 269 (1998), pp. 307–329].

Publication
SIAM Journal on Matrix Analysis and Applications
Date
Tags
Citation
T. G. Kolda. Orthogonal Tensor Decompositions. SIAM Journal on Matrix Analysis and Applications, Vol. 23, No. 1, pp. 243-255, 2001. https://doi.org/10.1137/S0895479800368354

Keywords

tensor decomposition, singular value decomposition, principal components analysis, multidimensional arrays

BibTeX

@article{Ko01,  
author = {Tamara G. Kolda}, 
title = {Orthogonal Tensor Decompositions}, 
journal = {SIAM Journal on Matrix Analysis and Applications}, 
volume = {23}, 
number = {1}, 
pages = {243--255}, 
month = {July}, 
year = {2001},
doi = {10.1137/S0895479800368354},
}