Improved polytope volume calculations based on Hamiltonian Monte Carlo with boundary reflections and sweet arithmetics
Why this work is in the frame
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Bibliographic record
Abstract
Computing the volume of a high dimensional polytope is a fundamental problem in geometry, also connected to the calculation of densities of states in statistical physics, and a central building block of such algorithms is the method used to sample a target probability distribution. This paper studies Hamiltonian Monte Carlo (HMC) with reflections on the boundary ofdomain, providing an enhanced alternative to Hit-and-run (HAR) to sample a target distribution restricted to the polytope. We make three contributions. First, we provide a convergence bound, paving the way to more precise mixing time analysis. Second, we present a robust implementation based on multi-precision arithmetic, a mandatory ingredient to guarantee exact predicates and robust constructions. We however allow controlled failures to happen, introducing the Sweeten Exact Geometric Computing (SEGC) paradigm. Third, we use our HMC random walk to perform H-polytope volume calculations, using it as an alternative to HAR within the volume algorithm by Cousins and Vempala. The systematic tests conducted up to dimension n = 100 on the cube, the isotropic and the standard simplex show that HMC significantly outperforms HAR both in terms of accuracy and running time. Additional tests show that calculations may be handled up to dimension n = 500. These tests also establish that multiprecision is mandatory to avoid exits from the polytope.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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