Information geometry has emerged from the study of the invariant structure in families of probability distributions. This invariance uniquely determines a second-order symmetric tensor g and third-order symmetric tensor T in a manifold of probability distributions. A pair of these tensors (g, T) defines a Riemannian metric and a pair of affine connections which together preserve the metric. Information geometry involves studying a Riemannian manifold having a pair of dual affine connections. Such a structure also arises from an asymmetric divergence function and affine differential geometry. A dually flat Riemannian manifold is particularly useful for various applications, because a generalized Pythagorean theorem and projection theorem hold. The Wasserstein distance gives another important geometry on probability distributions, which is non-invariant but responsible for the metric properties of a sample space. I attempt to construct information geometry of the entropy-regularized Wasserstein distance.

信息几何是从概率分布族不变结构的研究中出现的。这种不变性唯一地确定了概率分布流形中的二阶对称张量g和三阶对称张量T。这些张量对 ( g, T ) 定义黎曼度量和一对仿射连接，它们一起保留度量。信息几何涉及研究具有一对对偶仿射连接的黎曼流形。这种结构也源于不对称散度函数和仿射微分几何。双平坦黎曼流形对于各种应用特别有用，因为广义毕达哥拉斯定理和投影定理成立。 Wasserstein 距离给出了概率分布的另一个重要几何形状，它是非不变的，但负责样本空间的度量属性。我尝试构建熵正则化 Wasserstein 距离的信息几何。