# 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].

Type
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},
}