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].
tensor decomposition, singular value decomposition, principal components analysis, multidimensional arrays
@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},
}