CONVEXITY PROPERTIES OF THE CONDITION NUMBER
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Bibliographic record
Abstract
We define in the space of n×m matrices of rank n, n ≤ m, the condition Riemannian\n\t\t\t\t structure as follows: For a given matrix A the tangent space at A is equipped with the Hermitian\n\t\t\t\t inner product obtained by multiplying the usual Frobenius inner product by the inverse of the\n\t\t\t\t square of the smallest singular value of A denoted σn(A). When this smallest singular value has\n\t\t\t\t multiplicity 1, the function A → log(σn(A)−2) is a convex function with respect to the condition\n\t\t\t\t Riemannian structure that is t → log(σn(A(t))−2) is convex, in the usual sense for any geodesic\n\t\t\t\t A(t). In a more abstract setting, a function α defined on a Riemannian manifold (M, , ) is said\n\t\t\t\t to be self-convex when log α(γ(t)) is convex for any geodesic in (M, α , ). Necessary and sufficient\n\t\t\t\t conditions for self-convexity are given when α is C2. When α(x) = d(x,N)−2, where d(x,N) is the\n\t\t\t\t distance from x to a C2 submanifold N ⊂Rj, we prove that α is self-convex when restricted to the\n\t\t\t\t largest open set of points x where there is a unique closest point in N to x. We also show, using\n\t\t\t\t this more general notion, that the square of the condition number A F /σn(A) is self-convex in\n\t\t\t\t projective space and the solution variety.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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 it