  We derive nonlinear acceleration methods based on the limited memory BFGS
(L-BFGS) update formula for accelerating iterative optimization methods of
alternating least squares (ALS) type applied to canonical polyadic (CP) and
Tucker tensor decompositions. Our approach starts from linear preconditioning
ideas that use linear transformations encoded by matrix multiplications, and
extends these ideas to the case of genuinely nonlinear preconditioning, where
the preconditioning operation involves fully nonlinear transformations. As
such, the ALS-type iterations are used as fully nonlinear preconditioners for
L-BFGS, or, equivalently, L-BFGS is used as a nonlinear accelerator for ALS.
Numerical results show that the resulting methods perform much better than
either stand-alone L-BFGS or stand-alone ALS, offering substantial improvements
in terms of time-to-solution and robustness over state-of-the-art methods for
large and noisy tensor problems, including previously described acceleration
methods based on nonlinear conjugate gradients and nonlinear GMRES. Our
approach provides a general L-BFGS-based acceleration mechanism for nonlinear
optimization.
