Chebyshev Approximation and the Global Geometry of Model Predictions
Bibliographic record
Abstract
Complex nonlinear models are typically ill conditioned or sloppy; their predictions are significantly affected by only a small subset of parameter combinations, and parameters are difficult to reconstruct from model behavior. Despite forming an important universality class and arising frequently in practice when performing a nonlinear fit to data, formal and systematic explanations of sloppiness are lacking. By unifying geometric interpretations of sloppiness with Chebyshev approximation theory, we rigorously explain sloppiness as a consequence of model smoothness. Our approach results in universal bounds on model predictions for classes of smooth models, capturing global geometric features that are intrinsic to their model manifolds, and characterizing a universality class of models. We illustrate this universality using three models from disparate fields (physics, chemistry, biology): exponential curves, reaction rates from an enzyme-catalyzed chemical reaction, and an epidemiology model of an infected population.
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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".