The Wasserstein-Fisher-Rao (WFR) distance on $S^{2}$ has recently been shown to coincide with a classical elastic distance between $S^{2}$-immersions in the theory of Riemannian shape analysis. While this correspondence holds in dimension $2$, the analogous statement fails in general on $S^{1}$ and, in the case of convex curves, it cannot be derived from existing two-dimensional arguments. In this paper, we establish that for convex absolutely continuous immersions of $S^{1}$ in the plane, the shape distance induced by the square root velocity transformation (SRVT) is indeed equivalent to the WFR distance acting on their associated length measures. The proof exploits a monotonicity principle for optimal transport on the universal cover of the circle, which in turn guarantees the existence of an optimal reparametrization achieving the SRVT infimum and enables a one-dimensional unbalanced optimal transport reformulation. Motivated by this equivalence, we further investigate the role of sparsity in shape optimization problems formulated in terms of length measures and regularized by the WFR distance. We study linear optimization over the corresponding balls, for which we prove a finiteness result when the reference measure is discrete, and propose a convex, positively one-homogeneous regularizer suitable for conditional gradient algorithms.
The Advective Fisher-Rao Geometry of Deterministic Measure Transport
Institut für Mathematik, Technische Universität Berlin（柏林工业大学数学研究所）；Max-Planck-Institut für Mathematik in den Naturwissenschaften（马克斯·普朗克数学研究所（自然科学领域））
A Fisher-Rao gradient flow for entropic mean-field min-max games
Heriot-Watt University（赫瑞瓦特大学）；Maxwell Institute for Mathematical Sciences（麦克斯韦数学研究所）；University of Edinburgh（爱丁堡大学）；The Alan Turing Institute（阿兰·图灵研究所）；Simtopia（Simtopia公司）
Center for Advanced Intelligence Project, RIKEN AIP, Japan（日本RIKEN AIP高级智能项目中心）；School of Mathematics, University of Edinburgh, UK（爱丁堡大学数学学院）；School of Mathematics, University of Edinburgh, UK, The Alan Turing Institute, UK（爱丁堡大学数学学院、英国艾伦·图灵研究所）
Fisher-Rao Gradient Flows of Linear Programs and State-Action Natural Policy Gradients
RWTH Aachen University（亚琛工业大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）
