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Mathematics > Optimization and Control

Title: Natural gradient via optimal transport I

Abstract: We study a natural Wasserstein gradient flow on manifolds of probability distributions with discrete sample spaces. We derive the Riemannian structure for the probability simplex from the dynamical formulation of the Wasserstein distance on a weighted graph. We pull back the geometric structure to the parameter space of any given probability model, which allows us to define a natural gradient flow there. In contrast to the natural Fisher-Rao gradient, the natural Wasserstein gradient incorporates a ground metric on sample space. We discuss implementations following the forward and backward Euler methods. We illustrate the analysis on elementary exponential family examples.
Subjects: Optimization and Control (math.OC); Information Theory (cs.IT)
Cite as: arXiv:1803.07033 [math.OC]
  (or arXiv:1803.07033v1 [math.OC] for this version)

Submission history

From: Wuchen Li [view email]
[v1] Fri, 16 Mar 2018 17:10:35 GMT (76kb,D)