Signed Graph Metric Learning via Gershgorin Disc Perfect Alignment
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Bibliographic record
Abstract
Given a convex and differentiable objective <inline-formula><tex-math notation="LaTeX">$Q({\mathbf M})$</tex-math></inline-formula> for a real symmetric matrix <inline-formula><tex-math notation="LaTeX">${\mathbf M}$</tex-math></inline-formula> in the positive definite (PD) cone—used to compute Mahalanobis distances—we propose a fast general metric learning framework that is entirely projection-free. We first assume that <inline-formula><tex-math notation="LaTeX">${\mathbf M}$</tex-math></inline-formula> resides in a space <inline-formula><tex-math notation="LaTeX">${\mathcal S}$</tex-math></inline-formula> of generalized graph Laplacian matrices corresponding to balanced signed graphs. <inline-formula><tex-math notation="LaTeX">${\mathbf M}\in {\mathcal S}$</tex-math></inline-formula> that is also PD is called a graph metric matrix. Unlike low-rank metric matrices common in the literature, <inline-formula><tex-math notation="LaTeX">${\mathcal S}$</tex-math></inline-formula> includes the important diagonal-only matrices as a special case. The key theorem to circumvent full eigen-decomposition and enable fast metric matrix optimization is Gershgorin disc perfect alignment (GDPA): given <inline-formula><tex-math notation="LaTeX">${\mathbf M}\in {\mathcal S}$</tex-math></inline-formula> and diagonal matrix <inline-formula><tex-math notation="LaTeX">${\mathbf S}$</tex-math></inline-formula> , where <inline-formula><tex-math notation="LaTeX">$S_{ii} = 1/v_i$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">${\mathbf v}$</tex-math></inline-formula> is the first eigenvector of <inline-formula><tex-math notation="LaTeX">${\mathbf M}$</tex-math></inline-formula> , we prove that Gershgorin disc left-ends of similarity transform <inline-formula><tex-math notation="LaTeX">${\mathbf B}= {\mathbf S}{\mathbf M}{\mathbf S}^{-1}$</tex-math></inline-formula> are perfectly aligned at the smallest eigenvalue <inline-formula><tex-math notation="LaTeX">$\lambda _{\min }$</tex-math></inline-formula> . Using this theorem, we replace the PD cone constraint in the metric learning problem with tightest possible linear constraints per iteration, so that the alternating optimization of the diagonal / off-diagonal terms in <inline-formula><tex-math notation="LaTeX">${\mathbf M}$</tex-math></inline-formula> can be solved efficiently as linear programs via the Frank-Wolfe method. We update <inline-formula><tex-math notation="LaTeX">${\mathbf v}$</tex-math></inline-formula> using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as entries in <inline-formula><tex-math notation="LaTeX">${\mathbf M}$</tex-math></inline-formula> are optimized successively. Experiments show that our graph metric optimization is significantly faster than cone-projection schemes, and produces competitive binary classification performance.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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