Do optimal entropy-constrained quantizers have a finite or infinite number of codewords?
Bibliographic record
Abstract
An entropy-constrained quantizer Q is optimal if it minimizes the expected distortion D(Q) subject to a constraint on the output entropy H(Q). We use the Lagrangian formulation to show the existence and study the structure of optimal entropy-constrained quantizers that achieve a point on the lower convex hull of the operational distortion-rate function D/sub h/(R) = inf/sub Q/{D(Q) : H(Q) /spl les/ R}. In general, an optimal entropy-constrained quantizer may have a countably infinite number of codewords. Our main results show that if the tail of the source distribution is sufficiently light (resp., heavy) with respect to the distortion measure, the Lagrangian-optimal entropy-constrained quantizer has a finite (resp., infinite) number of codewords. In particular, for the squared error distortion measure, if the tail of the source distribution is lighter than the tail of a Gaussian distribution, then the Lagrangian-optimal quantizer has only a finite number of codewords, while if the tail is heavier than that of the Gaussian, the Lagrangian-optimal quantizer has an infinite number of codewords.
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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 | 0.025 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.008 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".