Optimal transport natural gradient for statistical manifolds with continuous sample space

Abstract

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. When parametrized densities lie in $\bR$, the induced metric tensor establishes an explicit formula. In optimization problems, we observe that the natural gradient descent outperforms the standard gradient descent when the Wasserstein distance is the objective function. In such a case, we prove that the resulting algorithm behaves similarly to the Newton method in the asymptotic regime. The proof calculates the exact Hessian formula for the Wasserstein distance, which further motivates another preconditioner for the optimization process. To the end, we present examples to illustrate the effectiveness of the natural gradient in several parametric statistical models, including the Gaussian measure, Gaussian mixture, Gamma distribution, and Laplace distribution.

References in corpus (7)

• The quadratic Wasserstein metric for earthquake location
• Transport information geometry I: Riemannian calculus on probability simplex
• On parameter estimation with the Wasserstein distance
• Population games and Discrete optimal transport
• Wasserstein Riemannian Geometry of Positive Definite Matrices
• The Extended Kalman Filter is a Natural Gradient Descent in Trajectory Space
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Cited by in corpus (5)

• Statistical Inference for Generative Models with Maximum Mean Discrepancy
• On parameter estimation with the Wasserstein distance
• Pseudo-Riemannian geometry embeds information geometry in optimal transport
• Constrained dynamical optimal transport and its Lagrangian formulation
• Efficient Wasserstein Natural Gradients for Reinforcement Learning