1D smooth optimal transport

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

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

# 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()
plot OT 1D smooth

Out:

<matplotlib.legend.Legend object at 0x7fecc9bdbf50>
pl.figure(2, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, M, 'Cost matrix 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')
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()
OT matrix Sinkhorn

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|
  110|2.474136e-10|

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()
OT matrix Smooth OT KL reg.
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()
OT matrix Smooth OT l2 reg.

Total running time of the script: ( 0 minutes 0.831 seconds)

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