Improving on the Cut-Set Bound via Geometric Analysis of Typical Sets
Bibliographic record
Abstract
We consider the discrete memoryless symmetric primitive relay channel, where, a source$X$wants to send information to a destination$Y$with the help of a relay$Z$and the relay can communicate to the destination via an error-free digital link of rate$R_{0}$, while$Y$and$Z$are conditionally independent and identically distributed given$X$. We develop two new upper bounds on the capacity of this channel that are tighter than existing bounds, including the celebrated cut-set bound. Our approach significantly deviates from the standard information-theoretic approach for proving upper bounds on the capacity of multi-user channels. We build on the blowing-up lemma to analyze the probabilistic geometric relations between the typical sets of the$n$-letter random variables associated with a reliable code for communicating over this channel. These relations translate to new entropy inequalities between the$n$-letter random variables involved. As an application of our bounds, we study an open question posed by (Cover, 1987), namely, what is the minimum rate$R_{0}^{*}$needed for the$Z$–$Y$link in order for the capacity of the relay channel to be equal to that of the broadcast cut. We consider the special case when the$X$–$Y$and$X$–$Z$links are both binary symmetric channels. Our tighter bounds on the capacity of the relay channel immediately translate to tighter lower bounds for$R_{0}^{*}$. More interestingly, we show that when$p\to 1/2$,$R_{0}^{*}\geq 0.1803$; even though the broadcast channel becomes completely noisy as$p\to 1/2$and its capacity, and therefore the capacity of the relay channel, goes to zero, a strictly positive rate$R_{0}$is required for the relay channel capacity to be equal to the broadcast bound. Existing upper bounds on the capacity of the relay channel, and the cut-set bound in particular, would rather imply$R_{0}^{*}\to 0$, while achievability schemes require$R_{0}^{*}\to 1$. We conjecture that$R_{0}^{*}\to 1$as$p\to 1/2$.
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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 | 0.004 | 0.029 |
| Meta-epidemiology (narrow) | 0.003 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.004 | 0.004 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.004 | 0.011 |
| Open science | 0.005 | 0.007 |
| Research integrity | 0.003 | 0.006 |
| Insufficient payload (model declined to judge) | 0.017 | 0.004 |
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".