We derive a class of divergences measuring the difference between probability density functions on the one-dimensional sample space. This divergence is a one-parameter variation of the Itakura--Saito divergence between quantile density functions. We prove that the proposed divergence is a one-parameter variation of the transport Kullback-Leibler divergence and the Hessian distance of negative Boltzmann entropy with respect to the Wasserstein-$2$ metric. From Taylor expansions, we also formulate the $3$-symmetric tensor in Wasserstein-$2$ space, which is given by an iterative Gamma three operator. The alpha--geodesic on Wasserstein space is also derived. From these properties, we name the proposed divergences transport alpha divergences. We provide several examples of transport alpha divergences on one dimensional distributions, such as generative models and Cauchy distributions.

翻译：我们推导了一类用于衡量一维样本空间上概率密度函数差异的散度。该散度是分位数密度函数间Itakura--Saito散度的单参数变体。我们证明了所提出的散度是传输Kullback-Leibler散度的单参数变体，同时也是负玻尔兹曼熵关于Wasserstein-$2$度量的Hessian距离。通过泰勒展开，我们还构建了Wasserstein-$2$空间中的$3$-对称张量，该张量由迭代Gamma三阶算子给出。同时推导了Wasserstein空间上的alpha--测地线。基于这些特性，我们将所提出的散度命名为传输alpha散度。我们提供了若干一维分布上传输alpha散度的实例，例如生成模型和柯西分布。