Heat kernels on unit spheres and applications to graph kernels
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Bibliographic record
Abstract
It is known that many statistical and machine learning approaches heavily rely on pairwise distance between data points.The choice of distance function on the underlying manifold has a fundamental impact on performance of these processes.This is closely related to questions of how to appropriately calculate distances, and hence, fundamental solutions (heat kernels) for heat operators can be obtained.In general, it is not so easy to obtain a closed form for heat kernels.We first survey results of heat kernels on radially symmetric Riemannian manifolds, e.g., Euclidean spaces and unit spheres in R n .For the cases n = 1, 2, 3, we may construct the heat kernel explicitly.But, the computation is much more complicated when n > 3.However, by results of Nagase, we may construct parametrices for the heat kernel by using elementary functions so that the error terms can be under controlled.In the second part of the paper, we discuss some results on subRiemannian manifolds, especially 3-dimensional sphere in C 2 as a CR-manifold.We study geodesics connecting two given points on S 3 respecting the Hopf fibration.This geodesic boundary value problem is completely solved in the case of S 3 and some partial results are obtained in the general case.The Carnot-Carathéodory distance is calculated.We also present some motivations related to quantum mechanics.Then we give a brief discussion of Greiner's methods on the heat kernel for the Cauchy-Riemann subLaplacian on S 2n+1 .We provide a brief discussion on applications of these heat kernels to graph kernels in the last part of the paper.
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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