Fast Interpolation-Based Globality Certificates for Computing Kreiss Constants and the Distance to Uncontrollability
Bibliographic record
Abstract
We propose a new approach to computing global minimizers of singular value functions in two real variables. Specifically, we present new algorithms to compute the Kreiss constant of a matrix and the distance to uncontrollability of a linear control system, both to arbitrary accuracy. Previous state-of-the-art methods for these two quantities rely on 2D level-set tests that are based on solving large eigenvalue problems. Consequently, these methods are costly, i.e., $\mathcal{O}(n^6)$ work using dense eigensolvers, and often multiple tests are needed before convergence. Divide-and-conquer techniques have been proposed that reduce the work complexity to $\mathcal{O}(n^4)$ on average and $\mathcal{O}(n^5)$ in the worst case, but these variants are nevertheless still very expensive and can be numerically unreliable. In contrast, our new interpolation-based globality certificates perform level-set tests by building interpolant approximations to certain one-variable continuous functions that are both relatively cheap and numerically robust to evaluate. Our new approach has an $\mathcal{O}(kn^3)$ work complexity and uses $\mathcal{O}(n^2)$ memory, where $k$ is the number of function evaluations necessary to build the interpolants. Not only is this interpolation process mostly “embarrassingly parallel," but also low-fidelity approximations typically suffice for all but the final interpolant, which must be built to high accuracy. Even without taking advantage of the aforementioned parallelism, $k$ is sufficiently small that our new approach is generally orders of magnitude faster than the previous state of the art.
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".