# Generalized Wasserstein Barycenter Demo

This example illustrates the computation of Generalized Wasserstein Barycenter as proposed in [42].

[42] Delon, J., Gozlan, N., and Saint-Dizier, A.. Generalized Wasserstein barycenters between probability measures living on different subspaces. arXiv preprint arXiv:2105.09755, 2021.

```# Author: Eloi Tanguy <eloi.tanguy@polytechnique.edu>
#

# sphinx_gallery_thumbnail_number = 2

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.pylab as pl
import ot
import matplotlib.animation as animation
```

## Generate and plot data

```# Input measures
sub_sample_factor = 8

sz = I1.shape[0]
UU, VV = np.meshgrid(np.arange(sz), np.arange(sz))

# Input measure locations in their respective 2D spaces
X_list = [np.stack((UU[im == 0], VV[im == 0]), 1) * 1.0 for im in [I1, I2, I3]]

# Input measure weights
a_list = [ot.unif(x.shape[0]) for x in X_list]

# Projections 3D -> 2D
P1 = np.array([[1, 0, 0], [0, 1, 0]])
P2 = np.array([[0, 1, 0], [0, 0, 1]])
P3 = np.array([[1, 0, 0], [0, 0, 1]])
P_list = [P1, P2, P3]

# Barycenter weights
weights = np.array([1 / 3, 1 / 3, 1 / 3])

# Number of barycenter points to compute
n_samples_bary = 150

# Send the input measures into 3D space for visualization
X_visu = [Xi @ Pi for (Xi, Pi) in zip(X_list, P_list)]

# Plot the input data
fig = plt.figure(figsize=(3, 3))
axis = fig.add_subplot(1, 1, 1, projection="3d")
for Xi in X_visu:
axis.scatter(Xi[:, 0], Xi[:, 1], Xi[:, 2], marker='o', alpha=.6)
axis.view_init(azim=45)
axis.set_xticks([])
axis.set_yticks([])
axis.set_zticks([])
plt.show()
```

## Barycenter computation and plot

```Y = ot.lp.generalized_free_support_barycenter(X_list, a_list, P_list, n_samples_bary)
fig = plt.figure(figsize=(3, 3))

axis = fig.add_subplot(1, 1, 1, projection="3d")
for Xi in X_visu:
axis.scatter(Xi[:, 0], Xi[:, 1], Xi[:, 2], marker='o', alpha=.6)
axis.scatter(Y[:, 0], Y[:, 1], Y[:, 2], marker='o', alpha=.6)
axis.view_init(azim=45)
axis.set_xticks([])
axis.set_yticks([])
axis.set_zticks([])
plt.show()
```

## Plotting projection matches

```fig = plt.figure(figsize=(9, 3))

ax = fig.add_subplot(1, 3, 1, projection='3d')
for Xi in X_visu:
ax.scatter(Xi[:, 0], Xi[:, 1], Xi[:, 2], marker='o', alpha=.6)
ax.scatter(Y[:, 0], Y[:, 1], Y[:, 2], marker='o', alpha=.6)
ax.view_init(elev=0, azim=0)
ax.set_xticks([])
ax.set_yticks([])
ax.set_zticks([])

ax = fig.add_subplot(1, 3, 2, projection='3d')
for Xi in X_visu:
ax.scatter(Xi[:, 0], Xi[:, 1], Xi[:, 2], marker='o', alpha=.6)
ax.scatter(Y[:, 0], Y[:, 1], Y[:, 2], marker='o', alpha=.6)
ax.view_init(elev=0, azim=90)
ax.set_xticks([])
ax.set_yticks([])
ax.set_zticks([])

ax = fig.add_subplot(1, 3, 3, projection='3d')
for Xi in X_visu:
ax.scatter(Xi[:, 0], Xi[:, 1], Xi[:, 2], marker='o', alpha=.6)
ax.scatter(Y[:, 0], Y[:, 1], Y[:, 2], marker='o', alpha=.6)
ax.view_init(elev=90, azim=0)
ax.set_xticks([])
ax.set_yticks([])
ax.set_zticks([])

plt.tight_layout()
plt.show()
```

## Rotation animation

```fig = plt.figure(figsize=(7, 7))
ax = fig.add_subplot(1, 1, 1, projection="3d")

def _init():
for Xi in X_visu:
ax.scatter(Xi[:, 0], Xi[:, 1], Xi[:, 2], marker='o', alpha=.6)
ax.scatter(Y[:, 0], Y[:, 1], Y[:, 2], marker='o', alpha=.6)
ax.view_init(elev=0, azim=0)
ax.set_xticks([])
ax.set_yticks([])
ax.set_zticks([])
return fig,

def _update_plot(i):
if i < 45:
ax.view_init(elev=0, azim=4 * i)
else:
ax.view_init(elev=i - 45, azim=4 * i)
return fig,

ani = animation.FuncAnimation(fig, _update_plot, init_func=_init, frames=136, interval=50, blit=True, repeat_delay=2000)
```