Bibliographic record
Abstract
Least-squares approximation is one of the most important methods for recovering an unknown function from data. While in many applications the data is fixed, in many others there is substantial freedom to choose where to sample. In this paper, we review recent progress on near-optimal random sampling strategies for (weighted) least-squares approximation in arbitrary linear spaces. We introduce the Christoffel function as a key quantity in the analysis of (weighted) least-squares approximation from random samples, then show how it can be used to construct a random sampling strategy, termed Christoffel sampling, that possesses near-optimal sample complexity: namely, the number of samples scales log-linearly in the dimension of the approximation space $n$. We discuss a series of variations, extensions and further topics, and throughout highlight connections to approximation theory, machine learning, information-based complexity and numerical linear algebra. Finally, motivated by various contemporary applications, we consider a generalization of the classical setting where the samples need not be pointwise samples of a scalar-valued function, and the approximation space need not be linear. We show that, even in this significantly more general setting, suitable generalizations of Christoffel function still determine the sample complexity. Consequently, these can be used to design enhanced, Christoffel sampling strategies in a unified way for general recovery problems. This article is largely self-contained, and intended to be accessible to nonspecialists.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.014 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".