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• POT Python Optimal Transport

Stochastic examples — POT Python Optimal Transport 0.9.5 documentation

This example is designed to show how to use the stochastic optimization algorithms for discrete and semi-continuous measures from the POT library.

[18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. Stochastic Optimization for Large-scale Optimal Transport. Advances in Neural Information Processing Systems (2016).

[19] Seguy, V., Bhushan Damodaran, B., Flamary, R., Courty, N., Rolet, A. & Blondel, M. Large-scale Optimal Transport and Mapping Estimation. International Conference on Learning Representation (2018)

# Author: Kilian Fatras <kilian.fatras@gmail.com>
#
# License: MIT License

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

Sample two discrete measures for the discrete case and compute their cost matrix c.

Call the “SAG” method to find the transportation matrix in the discrete case

[[2.55553509e-02 9.96395660e-02 1.76579142e-02 4.31178196e-06]
 [1.21640234e-01 1.25357448e-02 1.30225078e-03 7.37891338e-03]
 [3.56123975e-03 7.61451746e-02 6.31505947e-02 1.33831456e-07]
 [2.61515202e-02 3.34246014e-02 8.28734709e-02 4.07550428e-04]
 [9.85500870e-03 7.52288517e-04 1.08262628e-02 1.21423583e-01]
 [2.16904253e-02 9.03825797e-04 1.87178503e-03 1.18391107e-01]
 [4.15462212e-02 2.65987989e-02 7.23177216e-02 2.39440107e-03]]

Sample one general measure a, one discrete measures b for the semicontinuous case, the points where source and target measures are defined and compute the cost matrix.

Call the “ASGD” method to find the transportation matrix in the semicontinuous case.

[3.76510592 7.64094845 3.78917596 2.57007572 1.65543745 3.4893295
 2.70623359] [-2.50319213 -2.25852474 -0.82688144  5.5885983 ]
[[2.19802712e-02 1.03838786e-01 1.70349712e-02 3.11402024e-06]
 [1.20269164e-01 1.50177118e-02 1.44418382e-03 6.12608330e-03]
 [3.05271739e-03 7.90868636e-02 6.07174656e-02 9.63289956e-08]
 [2.33574229e-02 3.61718564e-02 8.30222147e-02 3.05648858e-04]
 [1.12749105e-02 1.04283861e-03 1.38926617e-02 1.16646732e-01]
 [2.49295484e-02 1.25865775e-03 2.41297662e-03 1.14255960e-01]
 [3.78279732e-02 2.93440562e-02 7.38545201e-02 1.83059335e-03]]

Compare the results with the Sinkhorn algorithm

[[2.55553508e-02 9.96395661e-02 1.76579142e-02 4.31178193e-06]
 [1.21640234e-01 1.25357448e-02 1.30225079e-03 7.37891333e-03]
 [3.56123974e-03 7.61451746e-02 6.31505947e-02 1.33831455e-07]
 [2.61515201e-02 3.34246014e-02 8.28734709e-02 4.07550425e-04]
 [9.85500876e-03 7.52288523e-04 1.08262629e-02 1.21423583e-01]
 [2.16904255e-02 9.03825804e-04 1.87178504e-03 1.18391107e-01]
 [4.15462212e-02 2.65987989e-02 7.23177217e-02 2.39440105e-03]]

For SAG

For ASGD

For Sinkhorn

Sample one general measure a, one discrete measures b for the semi-continuous case and compute the cost matrix c.

Call the “SGD” dual method to find the transportation matrix in the semi-continuous case

[0.91732819 2.7799397  1.07406199 0.01970121 0.60717156 1.80910257
 0.10902398] [0.34639291 0.47463643 1.57482501 4.92047485]
[[2.20200322e-02 9.25938748e-02 1.09047347e-02 9.25518158e-08]
 [1.60917795e-02 1.78850969e-03 1.23469888e-04 2.43170724e-05]
 [3.49209980e-03 8.05271170e-02 4.43815515e-02 3.26915633e-09]
 [3.15043415e-02 4.34264205e-02 7.15531236e-02 1.22305749e-05]
 [6.82992713e-02 5.62286712e-03 5.37746045e-02 2.09630346e-02]
 [8.02712798e-02 3.60737409e-03 4.96463916e-03 1.09144850e-02]
 [4.86875958e-02 3.36173252e-02 6.07394894e-02 6.98997703e-05]]

Call the Sinkhorn algorithm from POT

[[2.55553508e-02 9.96395661e-02 1.76579142e-02 4.31178193e-06]
 [1.21640234e-01 1.25357448e-02 1.30225079e-03 7.37891333e-03]
 [3.56123974e-03 7.61451746e-02 6.31505947e-02 1.33831455e-07]
 [2.61515201e-02 3.34246014e-02 8.28734709e-02 4.07550425e-04]
 [9.85500876e-03 7.52288523e-04 1.08262629e-02 1.21423583e-01]
 [2.16904255e-02 9.03825804e-04 1.87178504e-03 1.18391107e-01]
 [4.15462212e-02 2.65987989e-02 7.23177217e-02 2.39440105e-03]]

For SGD

For Sinkhorn

Total running time of the script: (0 minutes 6.416 seconds)

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