Optimal transport natural gradient for statistical manifolds with continuous sample space

Understand

We study the Wasserstein natural gradient in parametric statistical models with continuous sample spaces.

• Our approach is to pull back the $L^2$-Wasserstein metric tensor in the probability density space to a parameter space, equipping the latter with a positive definite metric tensor, under which it becomes a Riemannian manifold, named the Wasserstein statistical manifold.
• In general, it is not a totally geodesic sub-manifold of the density space, and therefore its geodesics will differ from the Wasserstein geodesics, except for the well-known Gaussian distribution case, a fact which can also be validated under our framework.
• We use the sub-manifold geometry to derive a gradient flow and natural gradient descent method in the parameter space.