Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon
Bibliographic record
Abstract
We introduce a method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we establish, provided the dimension of the reconstruction space is chosen suitably in relation to the number of samples, this procedure can be numerically implemented in a stable manner. Moreover, the accuracy of the resulting approximation is completely determined by the choice of reconstruction basis, meaning that the reconstruction vectors can be tailored to the particular problem at hand. An important example of this approach is the accurate recovery of a piecewise analytic function from its first few Fourier coefficients. Whilst the standard Fourier projection suffers from the Gibbs phenomenon, by reconstructing in a piecewise polynomial basis, we obtain an approximation with root exponential accuracy in terms of the number of Fourier samples and exponential accuracy in terms of the degree of the reconstruction function. Numerical examples illustrate the advantage of this approach over other existing methods.
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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 | 0.002 | 0.006 |
| 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 | 0.003 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".