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The Wasserstein Distance and Optimal Transport Map of Gaussian Processes

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WGPOT: Wasserstein Distance and Optimal Transport Map of Gaussian Processes

This repository contains a Python implementation of the Wasserstein Distance, Wasserstein Barycenter and Optimal Transport Map of Gaussian Processes. Based on the papers:

File Description

  • wgpot.py contains all functions for computing Wasserstein distance, Barycenter and transport map
  • example.py includes several simple examples
  • utils.py includes functions for data preprocessing and visualization

Requirement

  • Python 3
  • Numpy
  • scipy

Examples

  • Compute the Wasserstein distance between two Gaussian Processes
# Import the function
from wgpot import Wasserstein_GP

gp_0 = (mu_0, k_0)     
gp_1 = (mu_1, k_1)
# mu_0/mu_1 (ndarray (n, 1)) is the mean of one Gaussian Process 
# K_0/K_1 (ndarray (n, n)) is the covariance matrix of one 
# Gaussain Process

wd_gp = Wasserstein_GP(gp_0, gp_1)
  • Compute the Barycenter of a set of Gaussian Processes
# Import the functions
from wgpot import GP_W_barycenter, Wasserstein_GP

gp_list = [(gp_0, k_0), (gp_1, k_1), ..., (gp_m, k_m)]  
# gp_list is the list of tuples contains the mean and covariance 
# matrix of one Gaussian Process

mu_bc, k_bc = GP_W_barycenter(gp_list)
  • Transport map (Push forward) from one Gaussian Process to another
from wgpot import expmap

v_mu, v_T = logmap(mu_0, k_0, mu_1, k_1)
# The logarithmic map from Gaussian Distributions on the 
# Riemannian manifold with the Wasseerstein metric

q_mu, q_K = expmap(gp_1_mu, gp_1_K, v_mu_t, v_T_t)
# Exponential map on the Riemannian manifold. For more detials, 
# please refer to [Takatsu, Asuka. 2001]

Results

  • The Wasserstein Barycenter between a set of Gaussian Processes

  • The transport map (geodesic) between two Gaussian Processes

  • The transport map between two 2-D Gaussian Processes