Signed Graph Metric Learning via Gershgorin Disc Perfect Alignment
Bibliographic record
Abstract
Given a convex and differentiable objective$Q({\mathbf M})$for a real symmetric matrix${\mathbf M}$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${\mathbf M}$resides in a space${\mathcal S}$of generalized graph Laplacian matrices corresponding to balanced signed graphs.${\mathbf M}\in {\mathcal S}$that is also PD is called a graph metric matrix. Unlike low-rank metric matrices common in the literature,${\mathcal S}$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${\mathbf M}\in {\mathcal S}$and diagonal matrix${\mathbf S}$, where$S_{ii} = 1/v_i$and${\mathbf v}$is the first eigenvector of${\mathbf M}$, we prove that Gershgorin disc left-ends of similarity transform${\mathbf B}= {\mathbf S}{\mathbf M}{\mathbf S}^{-1}$are perfectly aligned at the smallest eigenvalue$\lambda _{\min }$. 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${\mathbf M}$can be solved efficiently as linear programs via the Frank-Wolfe method. We update${\mathbf v}$using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as entries in${\mathbf M}$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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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 | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 0.003 |
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".