论文标题
沿schr {Ö} dinger桥的熵曲率在零温度下
Entropic curvature on graphs along Schr{ö}dinger bridges at zero temperature
论文作者
论文摘要
Lott-Sturm-Villani在地球空间上的曲率理论已通过C. l {é} Onard扩展到离散的图形空间，通过替换W2-Wasserstein Geodesics通过Schr {Ö} Odinger Bridges在熵曲率的定义中取代[23，25，24]。作为一个了不起的事实，随着温度参数为零，这些schr {Ö} dinger桥由空间的地球化学支持。我们在离散图上分析此属性以在离散空间上达到熵曲率。我们的方法为多个图形空间示例提供了熵曲率的下限：晶格Z N具有计数度量，赋予了产品概率测量，圆，完整的图形，Bernoulli-Laplace模型的离散立方体。我们的一般结果还适用于本文未专门研究的大量图。与图形上的ERBAR-MAAS相反[27，10，11]，本文的熵曲率结果暗示了上述图上所述图表的新运输entropy不平等的新pr {é} kopa-leindler类型的不平等类型以及与精制浓度相关的新运输 - entropy不平等。例如，在离散的HyperCube {0，1} n上，对于Bernoulli Laplace模型，将达到新的W2-W1传输 - entropy不等式，无法通过尺寸n上的通常的诱导参数来得出。令人惊讶的是，我们的方法还改善了与Talagrand [38]相关的与所谓的凸壳法相关的弱传输 - 融合不平等现象（见[28，15]）。
Lott-Sturm-Villani theory of curvature on geodesic spaces has been extended to discrete graph spaces by C. L{é}onard by replacing W2-Wasserstein geodesics by Schr{ö}odinger bridges in the definition of entropic curvature [23, 25, 24]. As a remarkable fact, as a temperature parameter goes to zero, these Schr{ö}dinger bridges are supported by geodesics of the space. We analyse this property on discrete graphs to reach entropic curvature on discrete spaces. Our approach provides lower bounds for the entropic curvature for several examples of graph spaces: the lattice Z n endowed with the counting measure, the discrete cube endowed with product probability measures, the circle, the complete graph, the Bernoulli-Laplace model. Our general results also apply to a large class of graphs which are not specifically studied in this paper. As opposed to Erbar-Maas results on graphs [27, 10, 11], entropic curvature results of this paper imply new Pr{é}kopa-Leindler type of inequalities on discrete spaces, and new transport-entropy inequalities related to refined concentration properties for the graphs mentioned above. For example on the discrete hypercube {0, 1} n and for the Bernoulli Laplace model, a new W2 -- W1 transport-entropy inequality is reached, that can not be derived by usual induction arguments over the dimension n. As a surprising fact, our method also gives improvements of weak transport-entropy inequalities (see [28, 15]) associated to the so-called convex-hull method by Talagrand [38].
