On the structure of optimal entropy-constrained scalar quantizers
Bibliographic record
Abstract
The nearest neighbor condition implies that when searching for a mean-square optimal fixed-rate quantizer it is enough to consider the class of regular quantizers, i.e., quantizers having convex cells and codepoints which lie inside the associated cells. In contrast, quantizer regularity can preclude optimality in entropy-constrained quantization. This can be seen by exhibiting a simple discrete scalar source for which the mean-square optimal entropy-constrained scalar quantizer (ECSQ) has disconnected (and hence nonconvex) cells at certain rates. In this work, new results concerning the structure and existence of optimal ECSQs are presented. One main result shows that for continuous sources and distortion measures of the form d(x,y)=/spl rho/(|x-y|), where /spl rho/ is a nondecreasing convex function, any finite-level ECSQ can be "regularized" so that the resulting regular quantizer has the same entropy and equal or less distortion. Regarding the existence of optimal ECSQs, we prove that under rather general conditions there exists an "almost regular" optimal ECSQ for any entropy constraint. For the squared error distortion measure and sources with piecewise-monotone and continuous densities, the existence of a regular optimal ECSQ is shown.
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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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".