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

TL;DR

This paper introduces a nonlinear acceleration technique combining L-BFGS with ALS for tensor decompositions, significantly improving efficiency and robustness over existing methods in large, noisy problems.

Contribution

It develops a novel nonlinear preconditioning approach that enhances ALS with L-BFGS, outperforming traditional methods in tensor decomposition tasks.

Findings

Methods outperform stand-alone L-BFGS and ALS.

Significant reduction in time-to-solution.

Enhanced robustness in noisy tensor problems.

Abstract

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.