By analogy to the terminology of curved exponential families in statistics, we define curved Bregman divergences as Bregman divergences restricted to non-affine parameter subspaces and sub-dimensional Bregman divergences when the restrictions are affine. A common example of curved Bregman divergence is the cosine dissimilarity between normalized vectors: a curved squared Euclidean divergence. We prove that the barycenter of a finite weighted set of parameters under a curved Bregman divergence amounts to the right Bregman projection onto the non-affine subspace of the barycenter with respect to the full Bregman divergence, and interpret a generalization of the weighted Bregman centroid of $n$ parameters as a $n$-fold sub-dimensional Bregman divergence. We demonstrate the significance of curved Bregman divergences with several examples: (1) symmetrized Bregman divergences, (2) pointwise symmetrized Bregman divergences, and (3) the Kullback-Leibler divergence between circular complex normal distributions. We explain how to reparameterize sub-dimensional Bregman divergences on simplicial sub-dimensional domains. We then consider monotonic embeddings to define representational curved Bregman divergences and show that the $α$-divergences are representational curved Bregman divergences with respect to $α$-embeddings of the probability simplex into the positive measure cone. As an application, we report an efficient method to calculate the intersection of a finite set of $α$-divergence spheres. As an application, we report an efficient method to calculate the intersection of a finite set of $α$-divergence spheres.

翻译：类比统计学中曲率指数族的概念，我们将布雷格曼散度限制在非仿射参数子空间上时定义为曲率布雷格曼散度，而当限制为仿射时则称为子维布雷格曼散度。曲率布雷格曼散度的一个常见例子是归一化向量间的余弦相异度：一种曲率平方欧几里得散度。我们证明，在曲率布雷格曼散度下，有限加权参数集的质心等价于在全布雷格曼散度下向该质心所在非仿射子空间的右布雷格曼投影，并将 $n$ 个参数的加权布雷格曼质心的推广解释为 $n$ 重子维布雷格曼散度。我们通过若干实例阐明曲率布雷格曼散度的重要性：(1) 对称化布雷格曼散度，(2) 逐点对称化布雷格曼散度，以及 (3) 圆复正态分布间的 Kullback-Leibler 散度。我们阐释了如何在单纯形子维域上对子维布雷格曼散度进行重参数化。随后通过引入单调嵌入来定义表示性曲率布雷格曼散度，并证明 $α$-散度正是概率单纯形通过 $α$-嵌入映射到正测度锥时对应的表示性曲率布雷格曼散度。作为应用，我们提出了一种高效计算有限个 $α$-散度球交集的方法。