Premium
Conjugate gradient algorithms for best rank‐1 approximation of tensors
Author(s) -
Curtef O.,
Dirr G.,
Helmke U.
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700706
Subject(s) - tensor (intrinsic definition) , rank (graph theory) , conjugate gradient method , unitary state , quantum entanglement , mathematics , order (exchange) , approximation algorithm , algorithm , combinatorics , quantum , pure mathematics , quantum mechanics , physics , finance , political science , law , economics
Motivated by considerations of pure state entanglement in quantum information, we consider the problem of finding the best rank‐1 approximation to an arbitrary r ‐th order tensor. Reformulating the problem as an optimization problem on the Lie group SU ( n 1 ) ⊗ … ⊗ SU ( n r ) of so‐called local unitary transformations and exploiting its intrinsic geometry yields a new approach, which finally leads to Riemannian variant of the conjugate gradient algorithm. Numerical simulations support that our method offers an alternative to the higher‐order power method for computing the best rank‐1 approximation to a tensor. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)