Transport f divergences

We define a class of divergences to measure differences between probability density functions in one-dimensional sample space. The construction is based on the convex function with the Jacobi operator of mapping function that pushforwards one density to the other. We call these information measures {\em transport $f$-divergences}. We present several properties of transport $f$-divergences, including invariances, convexities, variational formulations, and Taylor expansions in terms of mapping functions. Examples of transport $f$-divergences in generative models are provided.