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

TL;DR

This paper introduces a deep learning framework to compute geodesics under the spherical Wasserstein-Fisher-Rao metric, enabling weighted sample generation and advancing understanding of this complex metric in applications like Bayesian inference.

Contribution

The paper develops a novel deep learning approach for geodesics under spherical WFR, incorporating a KL divergence term and a new regularization for improved sample generation.

Findings

Effective computation of spherical WFR geodesics using deep learning.

Enhanced weighted sample generation for Bayesian inference.

Improved understanding of geodesic structures in spherical WFR space.

Abstract

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.