We establish a dimension-free improvement of Talagrand’s Gaussian transport-entropy inequality, under the assumption that the measures satisfy a Poincaré inequality. We also study stability of the inequality, in terms of relative entropy, when restricted to measures whose covariance matrix trace is smaller than the ambient dimension. In case the covariance matrix is strictly smaller than the identity, we give dimension-free estimates which depend on its eigenvalues. To complement our results, we show that our conditions cannot be relaxed, and that there exist measures with covariance larger than the identity, for which the inequality is not stable, in relative entropy. To deal with these examples, we show that, without any assumptions, one can always get quantitative stability estimates in terms of relative entropy to Gaussian mixtures. Our approach gives rise to a new point of view which sheds light on the hierarchy between Fisher information, entropy and transportation distance, and may be of independent interest. In particular, it implies that the described results apply verbatim to the log-Sobolev inequality and improve upon some known estimates in the literature.

在测度满足Poincaré不等式的假设下，我们建立了Talagrand高斯输运熵不等式的无量纲改进。当限于协方差矩阵迹线小于环境维的度量时，我们还根据相对熵来研究不等式的稳定性。如果协方差矩阵严格小于恒等式，我们将根据其特征值给出无量纲估计。为了补充我们的结果，我们表明我们的条件不能放宽，并且存在不等式不稳定的相对熵的协方差大于恒等性的度量。为了处理这些示例，我们表明，在没有任何假设的情况下，总能获得相对于高斯混合的相对熵的定量稳定性估计。我们的方法提出了一种新观点，阐明了费舍尔信息，熵和运输距离之间的层次结构，并且可能具有独立的意义。特别地，这意味着所描述的结果逐字地应用于对数Sobolev不等式，并且在文献中的一些已知估计上有所改进。