Near-Optimal Sampling Strategies for Multivariate Function Approximation on General Domains
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Bibliographic record
Abstract
In this paper, we address the problem of approximating a multivariate function defined on a general domain in $d$ dimensions from sample points. We consider weighted least-squares approximation in an arbitrary finite-dimensional space $P$ from independent random samples taken according to a suitable measure. In general, least-squares approximations can be inaccurate and ill-conditioned when the number of sample points $M$ is close to $N = \dim(P)$. To counteract this, we introduce a novel method for sampling in general domains which leads to provably accurate and well-conditioned approximations. The resulting sampling measure is discrete and therefore straightforward to sample from. Our main result shows near-optimal sample complexity for this procedure; specifically, $M = \mathcal{O}(N \log(N))$ samples suffice for a well-conditioned and accurate approximation. Numerical experiments on polynomial approximation in general domains confirm the benefits of this method over standard sampling.
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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 | 0.006 | 0.007 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 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