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Most tensor problems are NP hard (2009)
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| Venue: | CORR |
| Citations: | 45 - 6 self |
BibTeX
@ARTICLE{Hillar09mosttensor,
author = {Christopher J. Hillar and Lek-heng Lim},
title = {Most tensor problems are NP hard},
journal = {CORR},
year = {2009}
}
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Abstract
The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has attracted a lot of attention recently. We examine here the computational tractability of some core problems in numerical multilinear algebra. We show that tensor analogues of several standard problems that are readily computable in the matrix (i.e. 2-tensor) case are NP hard. Our list here includes: determining the feasibility of a system of bilinear equations, determining an eigenvalue, a singular value, or the spectral norm of a 3-tensor, determining a best rank-1 approximation to a 3-tensor, determining the rank of a 3-tensor over R or C. Hence making tensor computations feasible is likely to be a challenge.
Keyphrases
tensor problem numerical multilinear algebra rank-1 approximation analogous collection computational tractability hypermatrices tensor engineering computing spectral norm core problem numerical linear algebra matrix computational method several standard problem tensor analogue singular value bilinear equation tensor computation feasible
