Channel capacity in the non-asymptotic regime: Taylor-type expansion and computable benchmarks
Bibliographic record
Abstract
In this paper, the non-asymptotic counterpart of Shannon capacity is investigated for any discrete input memory-less channel with discrete or continuous output (DIMC). Given any block length n and word error probability ϵ, let Rn(ϵ) be the best channel coding rate achievable with the block length n subject to the error probability e. Based on the non-asymptotic equipartition properties (NEP) established recently by Yang and Meng, a quantity δt, n(ϵ) is first defined to measure the relative magnitude of error probability ϵ and block length n with respect to a given DIMC and an input distribution t. Then, by combining the non-asymptotic achievability and converse established recently by Yang and Meng via jar decoding, it is shown that, given n and ϵn(ϵ) has a "Taylor-type expansion" with respect to δt, n(ϵ), with the first two terms of the expansion being maxt[I (t; P)-δt, n(ϵ)] = I(t*;P)-δt*, n(ϵ) for some optimal distribution t*, and the third order term being O(δ2t*, n) + O(ln n/n). Finally, based on the Taylor-type expansion and the non-asymptotic converse, two easy to compute approximation formulas for Rn(ϵ) (dubbed “SO” and “NEP”) are provided. Numerical results show that both the SO and NEP approximation formulas provide reliable and accurate estimation, in contrast with the normal approximation which sometimes falls below achievable bounds and sometimes rises above converses. An important implication arising from the Taylor-type expansion of Rn(ϵ) is that in the practical non-asymptotic regime, the optimal marginal codeword symbol distribution is not necessarily a Shannon capacity achieving distribution.
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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 | 0.003 | 0.029 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| 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".