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We derive nonlinear acceleration methods based on the limited memory BFGS(L-BFGS) update formula for accelerating iterative optimization methods ofalternating least squares (ALS) type applied to canonical polyadic (CP) andTucker tensor decompositions. Our approach starts from linear preconditioningideas that use linear transformations encoded by matrix multiplications, andextends these ideas to the case of genuinely nonlinear preconditioning, wherethe preconditioning operation involves fully nonlinear transformations. Assuch, the ALS-type iterations are used as fully nonlinear preconditioners forL-BFGS, or, equivalently, L-BFGS is used as a nonlinear accelerator for ALS.Numerical results show that the resulting methods perform much better thaneither stand-alone L-BFGS or stand-alone ALS, offering substantial improvementsin terms of time-to-solution and robustness over state-of-the-art methods forlarge and noisy tensor problems, including previously described accelerationmethods based on nonlinear conjugate gradients and nonlinear GMRES. Ourapproach provides a general L-BFGS-based acceleration mechanism for nonlinearoptimization.

Nonlinearly Preconditioned L-BFGS as an Acceleration Mechanism for Alternating Least Squares, with Application to Tensor Decomposition