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 $$X,Y \in {\mathbb {R}}^d$$ X , Y ∈ R d , the deficit in the Shannon–Stam inequality is bounded from below by the expression $$\begin{aligned} C \left( \mathrm {D}\left( X||G\right) + \mathrm {D}\left( Y||G\right) \right) , \end{aligned}$$ C D X | | G + D Y | | G , where $$\mathrm {D}\left( \cdot ~ ||G\right) $$ D · | | G 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.

中文翻译：

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通过 Föllmer 过程的 Shannon-Stam 不等式的稳定性

我们在熵和运输距离方面证明了对数凹随机向量的 Shannon-Stam 不等式（也称为熵-幂不等式）的稳定性估计。特别地，我们以下列形式给出一般对数凹随机向量的第一个稳定性估计：对于对数凹随机向量 $$X,Y \in {\mathbb {R}}^d$$ X , Y ∈ R d ，Shannon-Stam 不等式中的赤字由表达式 $$\begin{aligned} C \left( \mathrm {D}\left( X||G\right) + \mathrm {D}\ left( Y||G\right) \right) , \end{aligned}$$ CDX | | G + DY | | G , 其中 $$\mathrm {D}\left( \cdot ~ ||G\right) $$ D · | | G 表示相对于标准高斯的相对熵，常数 C 仅取决于协方差结构以及 X 和 Y 的光谱间隙。在一致对数凹向量的情况下，我们的分析给出了无维数界限。我们的证明基于一种新方法，该方法使用随机控制理论中的熵最小化过程。