plot_Intro_OT.py

# coding: utf-8
"""
=============================================
Introduction to Optimal Transport with Python
=============================================

This example gives an introduction on how to use Optimal Transport in Python.

"""

# Author: Remi Flamary, Nicolas Courty, Aurelie Boisbunon
#
# License: MIT License
# sphinx_gallery_thumbnail_number = 1

##############################################################################
# POT Python Optimal Transport Toolbox
# ------------------------------------
#
# POT installation
# ```````````````````
#
# * Install with pip::
#
#     pip install pot
# * Install with conda::
#
#     conda install -c conda-forge pot
#
# Import the toolbox
# ```````````````````
#

import numpy as np  # always need it
import pylab as pl  # do the plots

import ot  # ot

import time

##############################################################################
# Getting help
# `````````````
#
# Online  documentation : `<https://pythonot.github.io/all.html>`_
#
# Or inline help:
#

help(ot.dist)


##############################################################################
# First OT Problem
# ----------------
#
# We will solve the Bakery/Cafés problem of transporting croissants from a
# number of Bakeries to Cafés in a City (in this case Manhattan). We did a
# quick google map search in Manhattan for bakeries and Cafés:
#
# .. image:: images/bak.png
#     :align: center
#     :alt: bakery-cafe-manhattan
#     :width: 600px
#     :height: 280px
#
# We extracted from this search their positions and generated fictional
# production and sale number (that both sum to the same value).
#
# We have acess to the position of Bakeries ``bakery_pos`` and their
# respective production ``bakery_prod`` which describe the source
# distribution. The Cafés where the croissants are sold are defined also by
# their position ``cafe_pos`` and ``cafe_prod``, and describe the target
# distribution. For fun we also provide a
# map ``Imap`` that will illustrate the position of these shops in the city.
#
#
# Now we load the data
#
#

data = np.load('../data/manhattan.npz')

bakery_pos = data['bakery_pos']
bakery_prod = data['bakery_prod']
cafe_pos = data['cafe_pos']
cafe_prod = data['cafe_prod']
Imap = data['Imap']

print('Bakery production: {}'.format(bakery_prod))
print('Cafe sale: {}'.format(cafe_prod))
print('Total croissants : {}'.format(cafe_prod.sum()))


##############################################################################
# Plotting bakeries in the city
# -----------------------------
#
# Next we plot the position of the bakeries and cafés on the map. The size of
# the circle is proportional to their production.
#

pl.figure(1, (7, 6))
pl.clf()
pl.imshow(Imap, interpolation='bilinear')  # plot the map
pl.scatter(bakery_pos[:, 0], bakery_pos[:, 1], s=bakery_prod, c='r', ec='k', label='Bakeries')
pl.scatter(cafe_pos[:, 0], cafe_pos[:, 1], s=cafe_prod, c='b', ec='k', label='Cafés')
pl.legend()
pl.title('Manhattan Bakeries and Cafés')


##############################################################################
# Cost matrix
# -----------
#
#
# We can now compute the cost matrix between the bakeries and the cafés, which
# will be the transport cost matrix. This can be done using the
# `ot.dist <https://pythonot.github.io/all.html#ot.dist>`_ function that
# defaults to squared Euclidean distance but can return other things such as
# cityblock (or Manhattan distance).
#

C = ot.dist(bakery_pos, cafe_pos)

labels = [str(i) for i in range(len(bakery_prod))]
f = pl.figure(2, (14, 7))
pl.clf()
pl.subplot(121)
pl.imshow(Imap, interpolation='bilinear')  # plot the map
for i in range(len(cafe_pos)):
    pl.text(cafe_pos[i, 0], cafe_pos[i, 1], labels[i], color='b',
            fontsize=14, fontweight='bold', ha='center', va='center')
for i in range(len(bakery_pos)):
    pl.text(bakery_pos[i, 0], bakery_pos[i, 1], labels[i], color='r',
            fontsize=14, fontweight='bold', ha='center', va='center')
pl.title('Manhattan Bakeries and Cafés')

ax = pl.subplot(122)
im = pl.imshow(C, cmap="coolwarm")
pl.title('Cost matrix')
cbar = pl.colorbar(im, ax=ax, shrink=0.5, use_gridspec=True)
cbar.ax.set_ylabel("cost", rotation=-90, va="bottom")

pl.xlabel('Cafés')
pl.ylabel('Bakeries')
pl.tight_layout()


##############################################################################
# The red cells in the matrix image show the bakeries and cafés that are
# further away, and thus more costly to transport from one to the other, while
# the blue ones show those that are very close to each other, with respect to
# the squared Euclidean distance.


##############################################################################
# Solving the OT problem with `ot.emd <https://pythonot.github.io/all.html#ot.emd>`_
# -----------------------------------------------------------------------------------

start = time.time()
ot_emd = ot.emd(bakery_prod, cafe_prod, C)
time_emd = time.time() - start

##############################################################################
# The function returns the transport matrix, which we can then visualize (next section).

##############################################################################
# Transportation plan vizualization
# `````````````````````````````````
#
# A good vizualization of the OT matrix in the 2D plane is to denote the
# transportation of mass between a Bakery and a Café by a line. This can easily
# be done with a double ``for`` loop.
#
# In order to make it more interpretable one can also use the ``alpha``
# parameter of plot and set it to ``alpha=G[i,j]/G.max()``.

# Plot the matrix and the map
f = pl.figure(3, (14, 7))
pl.clf()
pl.subplot(121)
pl.imshow(Imap, interpolation='bilinear')  # plot the map
for i in range(len(bakery_pos)):
    for j in range(len(cafe_pos)):
        pl.plot([bakery_pos[i, 0], cafe_pos[j, 0]], [bakery_pos[i, 1], cafe_pos[j, 1]],
                '-k', lw=3. * ot_emd[i, j] / ot_emd.max())
for i in range(len(cafe_pos)):
    pl.text(cafe_pos[i, 0], cafe_pos[i, 1], labels[i], color='b', fontsize=14,
            fontweight='bold', ha='center', va='center')
for i in range(len(bakery_pos)):
    pl.text(bakery_pos[i, 0], bakery_pos[i, 1], labels[i], color='r', fontsize=14,
            fontweight='bold', ha='center', va='center')
pl.title('Manhattan Bakeries and Cafés')

ax = pl.subplot(122)
im = pl.imshow(ot_emd)
for i in range(len(bakery_prod)):
    for j in range(len(cafe_prod)):
        text = ax.text(j, i, '{0:g}'.format(ot_emd[i, j]),
                       ha="center", va="center", color="w")
pl.title('Transport matrix')

pl.xlabel('Cafés')
pl.ylabel('Bakeries')
pl.tight_layout()

##############################################################################
# The transport matrix gives the number of croissants that can be transported
# from each bakery to each café. We can see that the bakeries only need to
# transport croissants to one or two cafés, the transport matrix being very
# sparse.

##############################################################################
# OT loss and dual variables
# --------------------------
#
# The resulting wasserstein loss loss is of the form:
#
# .. math::
#     W=\sum_{i,j}\gamma_{i,j}C_{i,j}
#
# where :math:`\gamma` is the optimal transport matrix.
#

W = np.sum(ot_emd * C)
print('Wasserstein loss (EMD) = {0:.2f}'.format(W))

##############################################################################
# Regularized OT with Sinkhorn
# ----------------------------
#
# The Sinkhorn algorithm is very simple to code. You can implement it directly
# using the following pseudo-code
#
# .. image:: images/sinkhorn.png
#     :align: center
#     :alt: Sinkhorn algorithm
#     :width: 440px
#     :height: 240px
#
# In this algorithm, :math:`\oslash` corresponds to the element-wise division.
#
# An alternative is to use the POT toolbox with
# `ot.sinkhorn <https://pythonot.github.io/all.html#ot.sinkhorn>`_
#
# Be careful of numerical problems. A good pre-processing for Sinkhorn is to
# divide the cost matrix ``C`` by its maximum value.

##############################################################################
# Algorithm
# `````````

# Compute Sinkhorn transport matrix from algorithm
reg = 0.1
K = np.exp(-C / C.max() / reg)
nit = 100
u = np.ones((len(bakery_prod), ))
for i in range(1, nit):
    v = cafe_prod / np.dot(K.T, u)
    u = bakery_prod / (np.dot(K, v))
ot_sink_algo = np.atleast_2d(u).T * (K * v.T)  # Equivalent to np.dot(np.diag(u), np.dot(K, np.diag(v)))

# Compute Sinkhorn transport matrix with POT
ot_sinkhorn = ot.sinkhorn(bakery_prod, cafe_prod, reg=reg, M=C / C.max())

# Difference between the 2
print('Difference between algo and ot.sinkhorn = {0:.2g}'.format(np.sum(np.power(ot_sink_algo - ot_sinkhorn, 2))))

##############################################################################
# Plot the matrix and the map
# ```````````````````````````

print('Min. of Sinkhorn\'s transport matrix = {0:.2g}'.format(np.min(ot_sinkhorn)))

f = pl.figure(4, (13, 6))
pl.clf()
pl.subplot(121)
pl.imshow(Imap, interpolation='bilinear')  # plot the map
for i in range(len(bakery_pos)):
    for j in range(len(cafe_pos)):
        pl.plot([bakery_pos[i, 0], cafe_pos[j, 0]],
                [bakery_pos[i, 1], cafe_pos[j, 1]],
                '-k', lw=3. * ot_sinkhorn[i, j] / ot_sinkhorn.max())
for i in range(len(cafe_pos)):
    pl.text(cafe_pos[i, 0], cafe_pos[i, 1], labels[i], color='b',
            fontsize=14, fontweight='bold', ha='center', va='center')
for i in range(len(bakery_pos)):
    pl.text(bakery_pos[i, 0], bakery_pos[i, 1], labels[i], color='r',
            fontsize=14, fontweight='bold', ha='center', va='center')
pl.title('Manhattan Bakeries and Cafés')

ax = pl.subplot(122)
im = pl.imshow(ot_sinkhorn)
for i in range(len(bakery_prod)):
    for j in range(len(cafe_prod)):
        text = ax.text(j, i, np.round(ot_sinkhorn[i, j], 1),
                       ha="center", va="center", color="w")
pl.title('Transport matrix')

pl.xlabel('Cafés')
pl.ylabel('Bakeries')
pl.tight_layout()


##############################################################################
# We notice right away that the matrix is not sparse at all with Sinkhorn,
# each bakery delivering croissants to all 5 cafés with that solution. Also,
# this solution gives a transport with fractions, which does not make sense
# in the case of croissants. This was not the case with EMD.

##############################################################################
# Varying the regularization parameter in Sinkhorn
# ````````````````````````````````````````````````
#

reg_parameter = np.logspace(-3, 0, 20)
W_sinkhorn_reg = np.zeros((len(reg_parameter), ))
time_sinkhorn_reg = np.zeros((len(reg_parameter), ))

f = pl.figure(5, (14, 5))
pl.clf()
max_ot = 100  # plot matrices with the same colorbar
for k in range(len(reg_parameter)):
    start = time.time()
    ot_sinkhorn = ot.sinkhorn(bakery_prod, cafe_prod, reg=reg_parameter[k], M=C / C.max())
    time_sinkhorn_reg[k] = time.time() - start

    if k % 4 == 0 and k > 0:  # we only plot a few
        ax = pl.subplot(1, 5, k / 4)
        im = pl.imshow(ot_sinkhorn, vmin=0, vmax=max_ot)
        pl.title('reg={0:.2g}'.format(reg_parameter[k]))
        pl.xlabel('Cafés')
        pl.ylabel('Bakeries')

    # Compute the Wasserstein loss for Sinkhorn, and compare with EMD
    W_sinkhorn_reg[k] = np.sum(ot_sinkhorn * C)
pl.tight_layout()


##############################################################################
# This series of graph shows that the solution of Sinkhorn starts with something
# very similar to EMD (although not sparse) for very small values of the
# regularization parameter, and tends to a more uniform solution as the
# regularization parameter increases.
#

##############################################################################
# Wasserstein loss and computational time
# ```````````````````````````````````````
#

# Plot the matrix and the map
f = pl.figure(6, (4, 4))
pl.clf()
pl.title("Comparison between Sinkhorn and EMD")

pl.plot(reg_parameter, W_sinkhorn_reg, 'o', label="Sinkhorn")
XLim = pl.xlim()
pl.plot(XLim, [W, W], '--k', label="EMD")
pl.legend()
pl.xlabel("reg")
pl.ylabel("Wasserstein loss")

##############################################################################
# In this last graph, we show the impact of the regularization parameter on
# the Wasserstein loss. We can see that higher
# values of ``reg`` leads to a much higher Wasserstein loss.
#
# The Wasserstein loss of EMD is displayed for
# comparison. The Wasserstein loss of Sinkhorn can be a little lower than that
# of EMD for low values of ``reg``, but it quickly gets much higher.
#