Fast Interpolation-Based Globality Certificates for Computing Kreiss Constants and the Distance to Uncontrollability
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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 a $\mathcal{O}(kn^3)$ 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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it