CONVEXITY PROPERTIES OF THE CONDITION NUMBER
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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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 | 0.002 | 0.013 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.014 | 0.002 |
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".