Real-Time Computation of Legendre-Sobolev Approximations
Bibliographic record
Abstract
The present work is motivated by the problem of mathematical handwriting recognition where symbols are represented as plane curves, (X(λ), Y(λ)) parameterized by arc length λ ϵ [0, L]. Earlier work has shown that approximating the coordinate functions as certain truncated orthogonal polynomial series yields fast and effective recognition. It has been previously shown how to compute Legendre series representation in real time, as the curve is being traced out. In this article we show how to compute Legendre-Sobolev series representation in real time. The idea is to numerically integrate the moments of the coordinate functions as the curve is being traced. We show how the Legendre-Sobolev coefficients may be constructed either from the Legendre series coefficients or directly from the moments. Computing via Legendre series coefficients requires two matrix vector products, while the direct method requires only one.
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How this classification was reachedexpand
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 | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".