Sigma-delta (/spl Sigma//spl Delta/) quantization and finite frames
Why this work is in the frame
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Bibliographic record
Abstract
The K-level Sigma-Delta (/spl Sigma//spl Delta/) scheme with step size /spl delta/ is introduced as a technique for quantizing finite frame expansions for /spl Ropf//sup d/. Error estimates for various quantized frame expansions are derived, and, in particular, it is shown that /spl Sigma//spl Delta/ quantization of a unit-norm finite frame expansion in /spl Ropf//sup d/ achieves approximation error where N is the frame size, and the frame variation /spl sigma/(F,p) is a quantity which reflects the dependence of the /spl Sigma//spl Delta/ scheme on the frame. Here /spl par//spl middot//spl par/ is the d-dimensional Euclidean 2-norm. Lower bounds and refined upper bounds are derived for certain specific cases. As a direct consequence of these error bounds one is able to bound the mean squared error (MSE) by an order of 1/N/sup 2/. When dealing with sufficiently redundant frame expansions, this represents a significant improvement over classical pulse-code modulation (PCM) quantization, which only has MSE of order 1/N under certain nonrigorous statistical assumptions. /spl Sigma//spl Delta/ also achieves the optimal MSE order for PCM with consistent reconstruction.
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.003 |
| Open science | 0.000 | 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