Stability of the Shannon–Stam inequality via the Föllmer process
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Bibliographic record
Abstract
Abstract We prove stability estimates for the Shannon–Stam inequality (also known as the entropy-power inequality) for log-concave random vectors in terms of entropy and transportation distance. In particular, we give the first stability estimate for general log-concave random vectors in the following form: for log-concave random vectors $$X,Y \in {\mathbb {R}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> , the deficit in the Shannon–Stam inequality is bounded from below by the expression $$\begin{aligned} C \left( \mathrm {D}\left( X||G\right) + \mathrm {D}\left( Y||G\right) \right) , \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>C</mml:mi> <mml:mfenced> <mml:mi>D</mml:mi> <mml:mfenced> <mml:mi>X</mml:mi> <mml:mo>|</mml:mo> <mml:mo>|</mml:mo> <mml:mi>G</mml:mi> </mml:mfenced> <mml:mo>+</mml:mo> <mml:mi>D</mml:mi> <mml:mfenced> <mml:mi>Y</mml:mi> <mml:mo>|</mml:mo> <mml:mo>|</mml:mo> <mml:mi>G</mml:mi> </mml:mfenced> </mml:mfenced> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where $$\mathrm {D}\left( \cdot ~ ||G\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mfenced> <mml:mo>·</mml:mo> <mml:mspace/> <mml:mo>|</mml:mo> <mml:mo>|</mml:mo> <mml:mi>G</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> denotes the relative entropy with respect to the standard Gaussian and the constant C depends only on the covariance structures and the spectral gaps of X and Y . In the case of uniformly log-concave vectors our analysis gives dimension-free bounds. Our proofs are based on a new approach which uses an entropy-minimizing process from stochastic control theory.
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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 | 0.002 | 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 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it