Approximation, Kernelization, and Entropy-Dissipation of Gradient Flows: from Wasserstein to Fisher-Rao

Jia-Jie Zhu, Alexander Mielke

Abstract:

Motivated by various machine learning applications, we present a principled investigation
of gradient flow dissipation geometry, emphasizing the Fisher-Rao type gradient flows and
the interplay with Wasserstein space. Using the dynamic Benamou-Brenier formulation, we
reveal a few precise connections between those flow dissipation geometries and commonly
used machine learning tools such as Stein flows, kernel discrepancies, and nonparametric
regression. In addition, we present analysis results in terms of Lojasiewicz type functional
inequalities, with an explicit threshold condition for a family of entropy dissipation along the
Fisher-Rao flows. Finally, we establish rigorous evolutionary Γ-convergence for the FisherRao type gradient flows obtained via regression, justifying the approximation beyond static
point-wise convergence.