# Smooth and sparse OT example

This example illustrates the computation of Smooth and Sparse (KL an L2 reg.) OT and sparsity-constrained OT, together with their visualizations.

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

# sphinx_gallery_thumbnail_number = 5

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()
```
```<matplotlib.legend.Legend object at 0x7fba8143fe20>
```
```pl.figure(2, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, M, 'Cost matrix M')
```
```(<Axes: >, <Axes: >, <Axes: >)
```

## Solve Smooth OT

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

pl.figure(3, 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(4, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, Gsm, 'OT matrix Smooth OT l2 reg.')

pl.show()
```
```lambd = 1e-1

max_nz = 2  # two non-zero entries are permitted per column of the OT plan
Gsc = ot.smooth.smooth_ot_dual(
a, b, M, lambd, reg_type='sparsity_constrained', max_nz=max_nz)
pl.figure(5, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, Gsc, 'Sparsity constrained OT matrix; k=2.')

pl.show()
```

Total running time of the script: (0 minutes 0.833 seconds)

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