Natural gradient in Wasserstein statistical manifold

05/22/2018

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by   Yifan Chen, et al.

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We study the Wasserstein natural gradient in parametric statistical models with continuous sample space. Our approach is to pull back the L^2-Wasserstein metric tensor in probability density space to parameter space, under which the parameter space become a Riemannian manifold, named the Wasserstein statistical manifold. The gradient flow and natural gradient descent method in parameter space are then derived. When parameterized densities lie in , we show the induced metric tensor establishes an explicit formula. Computationally, optimization problems can be accelerated by the proposed Wasserstein natural gradient descent, if the objective function is the Wasserstein distance. Examples are presented to demonstrate its effectiveness in several parametric statistical models.