# Optimal Transport between 2D empirical distributions

Illustration of 2D optimal transport between distributions that are weighted sum of Diracs. The OT matrix is plotted with the samples.

```# Author: Remi Flamary <remi.flamary@unice.fr>
#         Kilian Fatras <kilian.fatras@irisa.fr>
#

# sphinx_gallery_thumbnail_number = 4

import numpy as np
import matplotlib.pylab as pl
import ot
import ot.plot
```

## Generate data

```n = 50  # nb samples

mu_s = np.array([0, 0])
cov_s = np.array([[1, 0], [0, 1]])

mu_t = np.array([4, 4])
cov_t = np.array([[1, -.8], [-.8, 1]])

xs = ot.datasets.make_2D_samples_gauss(n, mu_s, cov_s)
xt = ot.datasets.make_2D_samples_gauss(n, mu_t, cov_t)

a, b = np.ones((n,)) / n, np.ones((n,)) / n  # uniform distribution on samples

# loss matrix
M = ot.dist(xs, xt)
```

## Plot data

```pl.figure(1)
pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
pl.legend(loc=0)
pl.title('Source and target distributions')

pl.figure(2)
pl.imshow(M, interpolation='nearest')
pl.title('Cost matrix M')
```
```Text(0.5, 1.0, 'Cost matrix M')
```

## Compute EMD

```G0 = ot.emd(a, b, M)

pl.figure(3)
pl.imshow(G0, interpolation='nearest')
pl.title('OT matrix G0')

pl.figure(4)
ot.plot.plot2D_samples_mat(xs, xt, G0, c=[.5, .5, 1])
pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
pl.legend(loc=0)
pl.title('OT matrix with samples')
```
```Text(0.5, 1.0, 'OT matrix with samples')
```

## Compute Sinkhorn

```# reg term
lambd = 1e-1

Gs = ot.sinkhorn(a, b, M, lambd)

pl.figure(5)
pl.imshow(Gs, interpolation='nearest')
pl.title('OT matrix sinkhorn')

pl.figure(6)
ot.plot.plot2D_samples_mat(xs, xt, Gs, color=[.5, .5, 1])
pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
pl.legend(loc=0)
pl.title('OT matrix Sinkhorn with samples')

pl.show()
```
```/home/circleci/project/ot/bregman/_sinkhorn.py:531: UserWarning: Sinkhorn did not converge. You might want to increase the number of iterations `numItermax` or the regularization parameter `reg`.
warnings.warn("Sinkhorn did not converge. You might want to "
```

## Empirical Sinkhorn

```# reg term
lambd = 1e-1

Ges = ot.bregman.empirical_sinkhorn(xs, xt, lambd)

pl.figure(7)
pl.imshow(Ges, interpolation='nearest')
pl.title('OT matrix empirical sinkhorn')

pl.figure(8)
ot.plot.plot2D_samples_mat(xs, xt, Ges, color=[.5, .5, 1])
pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
pl.legend(loc=0)
pl.title('OT matrix Sinkhorn from samples')

pl.show()
```

Total running time of the script: (0 minutes 2.655 seconds)

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