Universal Weak Variable-Length Source Coding on Countably Infinite Alphabets
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Bibliographic record
Abstract
Motivated from the fact that universal source coding on countably infinite alphabets (∞-alphabets) is not feasible, this work introduces the notion of “almost lossless source coding”. Analog to the weak variable-length source coding problem studied by Han (IEEE Trans. Inf. Theory, vol. 46, no. 4, pp. 1217-1226, Jul. 2000), almost lossless source coding aims at relaxing the lossless block-wise assumption to allow an average per-letter distortion that vanishes asymptotically as the block-length tends to infinity. In this setup, we show on one hand that Shannon entropy characterizes the minimum achievable rate (similarly to the case of finite alphabet sources) while on the other that almost lossless universal source coding becomes feasible for the family of finite-entropy stationary memoryless sources with ∞-alphabets. Furthermore, we study a stronger notion of almost lossless universality that demands uniform convergence of the average per-letter distortion to zero, where we establish a necessary and sufficient condition for the so-called family of “envelope distributions” to achieve it. Remarkably, this condition is the same necessary and sufficient condition needed for the existence of a strongly minimax (lossless) universal source code for the family of envelope distributions. Finally, we show that an almost lossless coding scheme offers faster rate of convergence for the (minimax) redundancy compared to the well-known information radius developed for the lossless case at the expense of tolerating a non-zero distortion that vanishes to zero as the block-length grows. This shows that even when lossless universality is feasible, an almost lossless scheme can offer different regimes on the rates of convergence of the (worst case) redundancy versus the (worst case) distortion.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.004 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.002 |
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