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Wasserstein-Fisher-Rao (WFR) distance is a family of metrics to gauge thediscrepancy of two Radon measures, which takes into account both transportationand weight change. Spherical WFR distance is a projected version of WFRdistance for probability measures so that the space of Radon measures equippedwith WFR can be viewed as metric cone over the space of probability measureswith spherical WFR. Compared to the case for Wasserstein distance, theunderstanding of geodesics under the spherical WFR is less clear and still anongoing research focus. In this paper, we develop a deep learning framework tocompute the geodesics under the spherical WFR metric, and the learned geodesicscan be adopted to generate weighted samples. Our approach is based on aBenamou-Brenier type dynamic formulation for spherical WFR. To overcome thedifficulty in enforcing the boundary constraint brought by the weight change, aKullback-Leibler (KL) divergence term based on the inverse map is introducedinto the cost function. Moreover, a new regularization term using the particlevelocity is introduced as a substitute for the Hamilton-Jacobi equation for thepotential in dynamic formula. When used for sample generation, our frameworkcan be beneficial for applications with given weighted samples, especially inthe Bayesian inference, compared to sample generation with previous flowmodels.

A deep learning framework for geodesics under spherical Wasserstein-Fisher-Rao metric and its application for weighted sample generation