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

arXiv: https://arxiv.org/abs/1803.08849 · HTML: https://mlsanity.com/paper/1803.08849/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Mar 2018.

## Built on by (most cited first, 2 of 2)

- [Closed-form geodesics and trust-region method to calculate Riemannian logarithms on Stiefel and its quotient manifolds](https://mlsanity.com/paper/2103.13327/) (2021)
- [A Tractable Two-Step Linear Mixing Model Solved with Second-Order Optimization for Spectral Unmixing under Variability](https://mlsanity.com/paper/2502.17212/) (2025)

## Abstract (arXiv)

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.