Transport alpha divergences

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.

💡 Research Summary

The paper introduces a novel family of divergences, called Transport α Divergences, for measuring the discrepancy between one‑dimensional probability density functions (PDFs). The construction starts from optimal transport (OT) theory: in one dimension the optimal transport map that pushes a source density q to a target density p is given by the monotone map T(x)=Qₚ(F_q(x)), where Qₚ and F_q are the quantile and cumulative distribution functions of p and q, respectively. The derivative of this map, T′(x)=Q′ₚ(F_q(x))/Q′_q(F_q(x)), is strictly positive and can be interpreted as the ratio of the quantile density functions (QDFs) of p and q.

Using this ratio, the authors define a one‑parameter family of scalar functions
(f_{T,\alpha}(z)=\begin{cases}\frac{1}{\alpha^{2}}\big(z^{\alpha}-\alpha\log z-1\big), & \alpha\neq0\