# Smooth optimal transport example

This example illustrates the computation of EMD, Sinkhorn and smooth OT plans and their visualization.

```# Author: Remi Flamary <remi.flamary@unice.fr>
#

# sphinx_gallery_thumbnail_number = 6

import numpy as np
import matplotlib.pylab as pl
import ot
import ot.plot
from ot.datasets import make_1D_gauss as gauss
```

## Generate data

```n = 100  # nb bins

# bin positions
x = np.arange(n, dtype=np.float64)

# Gaussian distributions
a = gauss(n, m=20, s=5)  # m= mean, s= std
b = gauss(n, m=60, s=10)

# loss matrix
M = ot.dist(x.reshape((n, 1)), x.reshape((n, 1)))
M /= M.max()
```

## Plot distributions and loss matrix

```pl.figure(1, figsize=(6.4, 3))
pl.plot(x, a, 'b', label='Source distribution')
pl.plot(x, b, 'r', label='Target distribution')
pl.legend()
```

Out:

```<matplotlib.legend.Legend object at 0x7f5dbdecaf10>
```
```pl.figure(2, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, M, 'Cost matrix M')
```

## Solve EMD

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

pl.figure(3, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, G0, 'OT matrix G0')
```

## Solve Sinkhorn

```lambd = 2e-3
Gs = ot.sinkhorn(a, b, M, lambd, verbose=True)

pl.figure(4, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, Gs, 'OT matrix Sinkhorn')

pl.show()
```

Out:

```It.  |Err
-------------------
0|2.821142e-01|
10|7.695268e-02|
20|1.112774e-02|
30|1.571553e-03|
40|2.218100e-04|
50|3.130527e-05|
60|4.418267e-06|
70|6.235716e-07|
80|8.800770e-08|
90|1.242095e-08|
100|1.753030e-09|
```

## Solve Smooth OT

```lambd = 2e-3
Gsm = ot.smooth.smooth_ot_dual(a, b, M, lambd, reg_type='kl')

pl.figure(5, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, Gsm, 'OT matrix Smooth OT KL reg.')

pl.show()
```
```lambd = 1e-1
Gsm = ot.smooth.smooth_ot_dual(a, b, M, lambd, reg_type='l2')

pl.figure(6, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, Gsm, 'OT matrix Smooth OT l2 reg.')

pl.show()
```

Total running time of the script: ( 0 minutes 0.955 seconds)

Gallery generated by Sphinx-Gallery