A generalized projective dynamic for solving extreme and interior eigenvalue problems
Bibliographic record
Abstract
In [18] (Golub and Liao), a continuous-time system which is based on the projective dynamic is proposed to solve some concave optimization problems (with the unit ball constraint) resulted from extreme and interior eigenvalue problems. The convergence inside the unit ball is established; however, neither further convergence result outside the unit ball nor the stability analysis is available. Moreover, preliminary numerical experience indicates that this method is sensitive to a parameter whose optimal value is still difficult to determine. After analyzing the stability of this dynamic, in this paper, we develop a generalized model and analyze the convergence of the new model both inside and outside the unit ball. The flow of the generalized model is proved to converge almost globally to some eigenvector corresponding to the smallest eigenvalue, and share many surprisingly analogous properties with the Rayleigh quotient gradient flow. Links of our generalized projective dynamical system with other related works are also discussed. The efficiency of our new model is both addressed in theory and verified in numerical testing.
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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 | 0.001 |
| 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) | 0.003 | 0.000 |
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".