Conjugate gradient algorithms for best rank-1 approximation of tensors

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 (n₁) ⊗ … ⊗ 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)