  Wasserstein-Fisher-Rao (WFR) distance is a family of metrics to gauge the
discrepancy of two Radon measures, which takes into account both transportation
and weight change. Spherical WFR distance is a projected version of WFR
distance for probability measures so that the space of Radon measures equipped
with WFR can be viewed as metric cone over the space of probability measures
with spherical WFR. Compared to the case for Wasserstein distance, the
understanding of geodesics under the spherical WFR is less clear and still an
ongoing research focus. In this paper, we develop a deep learning framework to
compute the geodesics under the spherical WFR metric, and the learned geodesics
can be adopted to generate weighted samples. Our approach is based on a
Benamou-Brenier type dynamic formulation for spherical WFR. To overcome the
difficulty in enforcing the boundary constraint brought by the weight change, a
Kullback-Leibler (KL) divergence term based on the inverse map is introduced
into the cost function. Moreover, a new regularization term using the particle
velocity is introduced as a substitute for the Hamilton-Jacobi equation for the
potential in dynamic formula. When used for sample generation, our framework
can be beneficial for applications with given weighted samples, especially in
the Bayesian inference, compared to sample generation with previous flow
models.
