Gradient Descent Meets Shift-and-Invert Preconditioning for Eigenvector Computation
Bibliographic record
Abstract
There has been a recent surge of interest in developing theoretically faster algorithms for leading eigenvector computation. The key to achieving faster convergence rates therein is to use the classic shift-and-invert preconditioning technique on top of power methods. The underlying problem then can be reduced to a series of linear system subproblems that can leverage fast approximate least squares solvers. Despite the simplicity of the power iterations as the base method, it may suffer from making limited progress towards solutions. In this work, we consider that the shift-and-invert preconditioning is paired with a new base method, namely gradient descent search. By virtue of the flexibility of setting step-sizes in gradient search processes, we expect the shift-and-inverted gradient descent solver can outperform the shift-and-inverted power methods. In particular, we present a novel convergence analysis for this new pairing that achieves a rate at ˜ O ( √ λ1 λ1−λp+1 ) , where λi represents the i -th largest eigenvalue of the given real symmetric matrix and p is the multiplicity of λ1 . Our experimental studies show that the proposed algorithm can be significantly faster than the shift-and-inverted power method in practice.
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.008 | 0.004 |
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".