Transport information Bregman divergences
This work is significant for researchers in optimal transport and information theory, offering new theoretical insights into divergences in probability spaces.
This paper investigates Bregman divergences within the $L^2$-Wasserstein metric space of probability densities. It specifically derives the transport Kullback-Leibler (KL) divergence from a Bregman divergence of negative Boltzmann-Shannon entropy and provides analytical formulas for one-dimensional densities and Gaussian families.
We study Bregman divergences in probability density space embedded with the $L^2$-Wasserstein metric. Several properties and dualities of transport Bregman divergences are provided. In particular, we derive the transport Kullback-Leibler (KL) divergence by a Bregman divergence of negative Boltzmann-Shannon entropy in $L^2$-Wasserstein space. We also derive analytical formulas and generalizations of transport KL divergence for one-dimensional probability densities and Gaussian families.