A linear programming approach to difference-of-convex piecewise linear approximation
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Bibliographic record
Abstract
We address the problem of finding continuous piecewise linear (CPWL) approximations of deterministic functions of any dimension that satisfy any predefined error-tolerance, while keeping the number of polytopes that partition the approximation domain low. Specifically, we focus on overcoming the major computational bottleneck of the CPWL Approximation Algorithm (CPWL-AA) that has been proposed in the recent literature. CPWL-AA uses the difference-of-convex CPWL representation to search CPWL approximations which can partition the approximation domain to have polytopes of any shape. A computational bottleneck of the method is to solve a mixed-integer linear program (MILP) in which the number of binary variables is large for many problems of practical interest. In this paper, we overcome this by introducing a method that obtains a high quality solution of the MILP by iteratively solving a linear program (LP). We further reduce the computational expense by developing a method that treats some constraints in the LP problem as lazy constraints. Through a computational study we demonstrate that the proposed methods substantially reduce the computation time of CPWL-AA, while maintaining high quality CPWL approximations. With this, we demonstrate that we can generate CPWL approximations that satisfy predefined error-tolerances on functions of up to five dimensions within reasonable solution times.
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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 | 0.009 | 0.008 |
| 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 | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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