Nonlinearly preconditioned L‐BFGS as an acceleration mechanism for alternating least squares with application to tensor decomposition
Bibliographic record
Abstract
Summary We derive nonlinear acceleration methods based on the limited‐memory Broyden–Fletcher–Goldfarb–Shanno (L‐BFGS) update formula for accelerating iterative optimization methods of alternating least squares (ALS) type applied to canonical polyadic 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 the nonlinear generalized minimal residual method. Our approach provides a general L‐BFGS‐based acceleration mechanism for nonlinear optimization.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".