Infinite-Dimensional Hamilton–Jacobi Equations for Statistical Inference on Sparse Graphs
Bibliographic record
Abstract
Abstract. We study the well-posedness of an infinite-dimensional Hamilton–Jacobi equation posed on the set of nonnegative measures and with a monotonic nonlinearity. Our results will be used in a companion work to propose a conjecture and prove partial results concerning the asymptotic mutual information in the assortative stochastic block model in the sparse regime. The equation we consider is naturally stated in terms of the Gateaux derivative of the solution, unlike previous works in which the derivative is usually of transport type. We introduce an approximating family of finite-dimensional Hamilton–Jacobi equations and use the monotonicity of the nonlinearity to show that no boundary condition needs to be prescribed to establish well-posedness. The solution to the infinite-dimensional Hamilton–Jacobi equation is then defined as the limit of these approximating solutions. In the special setting of a convex nonlinearity, we also provide a Hopf–Lax variational representation of the solution.
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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 | 0.009 | 0.060 |
| Meta-epidemiology (narrow) | 0.002 | 0.003 |
| Meta-epidemiology (broad) | 0.004 | 0.002 |
| Bibliometrics | 0.003 | 0.004 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.005 | 0.005 |
| Research integrity | 0.005 | 0.007 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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".