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Most tensor problems are NP hard (2009)

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by Christopher J. Hillar , Lek-heng Lim
Venue:CORR
Citations:45 - 6 self
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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   

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