Approches primales et duales pour l'énumération des chambres d'un arrangement d'hyperplans réel - Le rapport complet
Bibliographic record
Abstract
The hyperplane arrangement problem appears in various theoretical and applied mathematical contexts. This paper focuses on the enumeration of the chambers of an arrangement, a task that most often requires algebraic or numerical computation. Among the recent numerical methods, Rada and Černý's recursive algorithm outperforms previous approches, by relying on a specific tree structure and on linear optimization. This paper presents modifications and improvements to this algorithm. It also introduces a dual approach solely grounded on matroid circuits and its associated concepts of <i>stem vectors</i>, thus reducing or avoiding the need to solve linear optimization problems. Along the way, theoretical properties of arrangements, such as their cardinality and conditions for their symmetry, completeness and connectivity, as well as properties of their various stem vector sets, are recalled or proved with an analytic viewpoint. It is shown, in particular, that the set of the chambers of an affine arrangement is located between those of two related linear arrangements. This leads to compact forms of the algorithms, which solve less subproblems. The proposed methods have been implemented in Julia and their efficiency is assessed on various instances of arrangements; for the best of them, this efficiency manifests itself by speedup ratios in the range [1.72,22.87] with an average value of 9.10.
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How this classification was reachedexpand
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 | 0.005 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".