diff options
Diffstat (limited to 'docs/source')
48 files changed, 3253 insertions, 40 deletions
diff --git a/docs/source/all.rst b/docs/source/all.rst index 32930fd..c968aa1 100644 --- a/docs/source/all.rst +++ b/docs/source/all.rst @@ -43,7 +43,7 @@ ot.da .. automodule:: ot.da :members: - + ot.gpu -------- @@ -80,3 +80,9 @@ ot.stochastic .. automodule:: ot.stochastic :members: + +ot.unbalanced +------------- + +.. automodule:: ot.unbalanced + :members: diff --git a/docs/source/auto_examples/auto_examples_jupyter.zip b/docs/source/auto_examples/auto_examples_jupyter.zip Binary files differindex 88e1e9b..901195a 100644 --- a/docs/source/auto_examples/auto_examples_jupyter.zip +++ b/docs/source/auto_examples/auto_examples_jupyter.zip diff --git a/docs/source/auto_examples/auto_examples_python.zip b/docs/source/auto_examples/auto_examples_python.zip Binary files differindex 120a586..ded2613 100644 --- a/docs/source/auto_examples/auto_examples_python.zip +++ b/docs/source/auto_examples/auto_examples_python.zip diff --git a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_001.png b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_001.png Binary files differindex 2e93ed1..a5bded7 100644 --- a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_001.png +++ b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_001.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_002.png b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_002.png Binary files differindex d6db0ed..1d90c2d 100644 --- a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_002.png +++ b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_002.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_005.png b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_005.png Binary files differindex 9a215ab..ea6a405 100644 --- a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_005.png +++ b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_005.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_006.png b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_006.png Binary files differindex 81c4ddb..8bc46dc 100644 --- a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_006.png +++ b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_006.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_009.png b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_009.png Binary files differindex 892b2a2..56d18ef 100644 --- a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_009.png +++ b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_009.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_010.png b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_010.png Binary files differindex c53717f..5aef7d2 100644 --- a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_010.png +++ b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_010.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_013.png b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_013.png Binary files differnew file mode 100644 index 0000000..bb8bd7c --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_013.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_014.png b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_014.png Binary files differnew file mode 100644 index 0000000..30cec7b --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_OT_2D_samples_014.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_UOT_1D_001.png b/docs/source/auto_examples/images/sphx_glr_plot_UOT_1D_001.png Binary files differnew file mode 100644 index 0000000..69ef5b7 --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_UOT_1D_001.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_UOT_1D_002.png b/docs/source/auto_examples/images/sphx_glr_plot_UOT_1D_002.png Binary files differnew file mode 100644 index 0000000..0407e44 --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_UOT_1D_002.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_UOT_1D_006.png b/docs/source/auto_examples/images/sphx_glr_plot_UOT_1D_006.png Binary files differnew file mode 100644 index 0000000..f58d383 --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_UOT_1D_006.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_001.png b/docs/source/auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_001.png Binary files differnew file mode 100644 index 0000000..ec8c51e --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_001.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_003.png b/docs/source/auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_003.png Binary files differnew file mode 100644 index 0000000..89ab265 --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_003.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_005.png b/docs/source/auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_005.png Binary files differnew file mode 100644 index 0000000..c6c49cb --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_005.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_006.png b/docs/source/auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_006.png Binary files differnew file mode 100644 index 0000000..8870b10 --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_006.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_barycenter_fgw_001.png b/docs/source/auto_examples/images/sphx_glr_plot_barycenter_fgw_001.png Binary files differnew file mode 100644 index 0000000..77e1282 --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_barycenter_fgw_001.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_barycenter_fgw_002.png b/docs/source/auto_examples/images/sphx_glr_plot_barycenter_fgw_002.png Binary files differnew file mode 100644 index 0000000..ca6d7f8 --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_barycenter_fgw_002.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_fgw_004.png b/docs/source/auto_examples/images/sphx_glr_plot_fgw_004.png Binary files differnew file mode 100644 index 0000000..4e0df9f --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_fgw_004.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_fgw_010.png b/docs/source/auto_examples/images/sphx_glr_plot_fgw_010.png Binary files differnew file mode 100644 index 0000000..d0e36e8 --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_fgw_010.png diff --git a/docs/source/auto_examples/images/sphx_glr_plot_fgw_011.png b/docs/source/auto_examples/images/sphx_glr_plot_fgw_011.png Binary files differnew file mode 100644 index 0000000..6d7e630 --- /dev/null +++ b/docs/source/auto_examples/images/sphx_glr_plot_fgw_011.png diff --git a/docs/source/auto_examples/images/thumb/sphx_glr_plot_OT_2D_samples_thumb.png b/docs/source/auto_examples/images/thumb/sphx_glr_plot_OT_2D_samples_thumb.png Binary files differindex b9135dd..ae33588 100644 --- a/docs/source/auto_examples/images/thumb/sphx_glr_plot_OT_2D_samples_thumb.png +++ b/docs/source/auto_examples/images/thumb/sphx_glr_plot_OT_2D_samples_thumb.png diff --git a/docs/source/auto_examples/images/thumb/sphx_glr_plot_UOT_1D_thumb.png b/docs/source/auto_examples/images/thumb/sphx_glr_plot_UOT_1D_thumb.png Binary files differnew file mode 100644 index 0000000..1d048f2 --- /dev/null +++ b/docs/source/auto_examples/images/thumb/sphx_glr_plot_UOT_1D_thumb.png diff --git a/docs/source/auto_examples/images/thumb/sphx_glr_plot_UOT_barycenter_1D_thumb.png b/docs/source/auto_examples/images/thumb/sphx_glr_plot_UOT_barycenter_1D_thumb.png Binary files differnew file mode 100644 index 0000000..999f175 --- /dev/null +++ b/docs/source/auto_examples/images/thumb/sphx_glr_plot_UOT_barycenter_1D_thumb.png diff --git a/docs/source/auto_examples/images/thumb/sphx_glr_plot_barycenter_fgw_thumb.png b/docs/source/auto_examples/images/thumb/sphx_glr_plot_barycenter_fgw_thumb.png Binary files differnew file mode 100644 index 0000000..9c3244e --- /dev/null +++ b/docs/source/auto_examples/images/thumb/sphx_glr_plot_barycenter_fgw_thumb.png diff --git a/docs/source/auto_examples/images/thumb/sphx_glr_plot_fgw_thumb.png b/docs/source/auto_examples/images/thumb/sphx_glr_plot_fgw_thumb.png Binary files differnew file mode 100644 index 0000000..609339d --- /dev/null +++ b/docs/source/auto_examples/images/thumb/sphx_glr_plot_fgw_thumb.png diff --git a/docs/source/auto_examples/index.rst b/docs/source/auto_examples/index.rst index 17a9710..fe6702d 100644 --- a/docs/source/auto_examples/index.rst +++ b/docs/source/auto_examples/index.rst @@ -29,13 +29,13 @@ This is a gallery of all the POT example files. .. raw:: html - <div class="sphx-glr-thumbcontainer" tooltip="Illustrates the use of the generic solver for regularized OT with user-designed regularization ..."> + <div class="sphx-glr-thumbcontainer" tooltip="This example illustrates the computation of Unbalanced Optimal transport using a Kullback-Leibl..."> .. only:: html - .. figure:: /auto_examples/images/thumb/sphx_glr_plot_optim_OTreg_thumb.png + .. figure:: /auto_examples/images/thumb/sphx_glr_plot_UOT_1D_thumb.png - :ref:`sphx_glr_auto_examples_plot_optim_OTreg.py` + :ref:`sphx_glr_auto_examples_plot_UOT_1D.py` .. raw:: html @@ -45,17 +45,17 @@ This is a gallery of all the POT example files. .. toctree:: :hidden: - /auto_examples/plot_optim_OTreg + /auto_examples/plot_UOT_1D .. raw:: html - <div class="sphx-glr-thumbcontainer" tooltip="Illustration of 2D Wasserstein barycenters if discributions that are weighted sum of diracs."> + <div class="sphx-glr-thumbcontainer" tooltip="Illustrates the use of the generic solver for regularized OT with user-designed regularization ..."> .. only:: html - .. figure:: /auto_examples/images/thumb/sphx_glr_plot_free_support_barycenter_thumb.png + .. figure:: /auto_examples/images/thumb/sphx_glr_plot_optim_OTreg_thumb.png - :ref:`sphx_glr_auto_examples_plot_free_support_barycenter.py` + :ref:`sphx_glr_auto_examples_plot_optim_OTreg.py` .. raw:: html @@ -65,17 +65,17 @@ This is a gallery of all the POT example files. .. toctree:: :hidden: - /auto_examples/plot_free_support_barycenter + /auto_examples/plot_optim_OTreg .. raw:: html - <div class="sphx-glr-thumbcontainer" tooltip="This example illustrates the computation of EMD, Sinkhorn and smooth OT plans and their visuali..."> + <div class="sphx-glr-thumbcontainer" tooltip="Illustration of 2D Wasserstein barycenters if discributions that are weighted sum of diracs."> .. only:: html - .. figure:: /auto_examples/images/thumb/sphx_glr_plot_OT_1D_smooth_thumb.png + .. figure:: /auto_examples/images/thumb/sphx_glr_plot_free_support_barycenter_thumb.png - :ref:`sphx_glr_auto_examples_plot_OT_1D_smooth.py` + :ref:`sphx_glr_auto_examples_plot_free_support_barycenter.py` .. raw:: html @@ -85,17 +85,17 @@ This is a gallery of all the POT example files. .. toctree:: :hidden: - /auto_examples/plot_OT_1D_smooth + /auto_examples/plot_free_support_barycenter .. raw:: html - <div class="sphx-glr-thumbcontainer" tooltip="This example is designed to show how to use the Gromov-Wassertsein distance computation in POT...."> + <div class="sphx-glr-thumbcontainer" tooltip="This example illustrates the computation of EMD, Sinkhorn and smooth OT plans and their visuali..."> .. only:: html - .. figure:: /auto_examples/images/thumb/sphx_glr_plot_gromov_thumb.png + .. figure:: /auto_examples/images/thumb/sphx_glr_plot_OT_1D_smooth_thumb.png - :ref:`sphx_glr_auto_examples_plot_gromov.py` + :ref:`sphx_glr_auto_examples_plot_OT_1D_smooth.py` .. raw:: html @@ -105,17 +105,17 @@ This is a gallery of all the POT example files. .. toctree:: :hidden: - /auto_examples/plot_gromov + /auto_examples/plot_OT_1D_smooth .. raw:: html - <div class="sphx-glr-thumbcontainer" tooltip="Illustration of 2D optimal transport between discributions that are weighted sum of diracs. The..."> + <div class="sphx-glr-thumbcontainer" tooltip="This example is designed to show how to use the Gromov-Wassertsein distance computation in POT...."> .. only:: html - .. figure:: /auto_examples/images/thumb/sphx_glr_plot_OT_2D_samples_thumb.png + .. figure:: /auto_examples/images/thumb/sphx_glr_plot_gromov_thumb.png - :ref:`sphx_glr_auto_examples_plot_OT_2D_samples.py` + :ref:`sphx_glr_auto_examples_plot_gromov.py` .. raw:: html @@ -125,7 +125,7 @@ This is a gallery of all the POT example files. .. toctree:: :hidden: - /auto_examples/plot_OT_2D_samples + /auto_examples/plot_gromov .. raw:: html @@ -209,6 +209,26 @@ This is a gallery of all the POT example files. .. raw:: html + <div class="sphx-glr-thumbcontainer" tooltip="Illustration of 2D optimal transport between discributions that are weighted sum of diracs. The..."> + +.. only:: html + + .. figure:: /auto_examples/images/thumb/sphx_glr_plot_OT_2D_samples_thumb.png + + :ref:`sphx_glr_auto_examples_plot_OT_2D_samples.py` + +.. raw:: html + + </div> + + +.. toctree:: + :hidden: + + /auto_examples/plot_OT_2D_samples + +.. raw:: html + <div class="sphx-glr-thumbcontainer" tooltip="This example is designed to show how to use the stochatic optimization algorithms for descrete ..."> .. only:: html @@ -289,6 +309,26 @@ This is a gallery of all the POT example files. .. raw:: html + <div class="sphx-glr-thumbcontainer" tooltip="This example illustrates the computation of regularized Wassersyein Barycenter as proposed in [..."> + +.. only:: html + + .. figure:: /auto_examples/images/thumb/sphx_glr_plot_UOT_barycenter_1D_thumb.png + + :ref:`sphx_glr_auto_examples_plot_UOT_barycenter_1D.py` + +.. raw:: html + + </div> + + +.. toctree:: + :hidden: + + /auto_examples/plot_UOT_barycenter_1D + +.. raw:: html + <div class="sphx-glr-thumbcontainer" tooltip="This example presents how to use MappingTransport to estimate at the same time both the couplin..."> .. only:: html @@ -329,6 +369,26 @@ This is a gallery of all the POT example files. .. raw:: html + <div class="sphx-glr-thumbcontainer" tooltip="This example illustrates the computation of FGW for 1D measures[18]."> + +.. only:: html + + .. figure:: /auto_examples/images/thumb/sphx_glr_plot_fgw_thumb.png + + :ref:`sphx_glr_auto_examples_plot_fgw.py` + +.. raw:: html + + </div> + + +.. toctree:: + :hidden: + + /auto_examples/plot_fgw + +.. raw:: html + <div class="sphx-glr-thumbcontainer" tooltip="This example introduces a domain adaptation in a 2D setting and the 4 OTDA approaches currently..."> .. only:: html @@ -409,6 +469,26 @@ This is a gallery of all the POT example files. .. raw:: html + <div class="sphx-glr-thumbcontainer" tooltip="This example illustrates the computation barycenter of labeled graphs using FGW"> + +.. only:: html + + .. figure:: /auto_examples/images/thumb/sphx_glr_plot_barycenter_fgw_thumb.png + + :ref:`sphx_glr_auto_examples_plot_barycenter_fgw.py` + +.. raw:: html + + </div> + + +.. toctree:: + :hidden: + + /auto_examples/plot_barycenter_fgw + +.. raw:: html + <div class="sphx-glr-thumbcontainer" tooltip="This example is designed to show how to use the Gromov-Wasserstein distance computation in POT...."> .. only:: html diff --git a/docs/source/auto_examples/plot_OT_2D_samples.ipynb b/docs/source/auto_examples/plot_OT_2D_samples.ipynb index 26831f9..dad138b 100644 --- a/docs/source/auto_examples/plot_OT_2D_samples.ipynb +++ b/docs/source/auto_examples/plot_OT_2D_samples.ipynb @@ -26,7 +26,7 @@ }, "outputs": [], "source": [ - "# Author: Remi Flamary <remi.flamary@unice.fr>\n#\n# License: MIT License\n\nimport numpy as np\nimport matplotlib.pylab as pl\nimport ot\nimport ot.plot" + "# Author: Remi Flamary <remi.flamary@unice.fr>\n# Kilian Fatras <kilian.fatras@irisa.fr>\n#\n# License: MIT License\n\nimport numpy as np\nimport matplotlib.pylab as pl\nimport ot\nimport ot.plot" ] }, { @@ -100,6 +100,24 @@ "source": [ "#%% sinkhorn\n\n# reg term\nlambd = 1e-3\n\nGs = ot.sinkhorn(a, b, M, lambd)\n\npl.figure(5)\npl.imshow(Gs, interpolation='nearest')\npl.title('OT matrix sinkhorn')\n\npl.figure(6)\not.plot.plot2D_samples_mat(xs, xt, Gs, color=[.5, .5, 1])\npl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')\npl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')\npl.legend(loc=0)\npl.title('OT matrix Sinkhorn with samples')\n\npl.show()" ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Emprirical Sinkhorn\n----------------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#%% sinkhorn\n\n# reg term\nlambd = 1e-3\n\nGes = ot.bregman.empirical_sinkhorn(xs, xt, lambd)\n\npl.figure(7)\npl.imshow(Ges, interpolation='nearest')\npl.title('OT matrix empirical sinkhorn')\n\npl.figure(8)\not.plot.plot2D_samples_mat(xs, xt, Ges, color=[.5, .5, 1])\npl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')\npl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')\npl.legend(loc=0)\npl.title('OT matrix Sinkhorn from samples')\n\npl.show()" + ] } ], "metadata": { @@ -118,7 +136,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.6.5" + "version": "3.6.8" } }, "nbformat": 4, diff --git a/docs/source/auto_examples/plot_OT_2D_samples.py b/docs/source/auto_examples/plot_OT_2D_samples.py index bb952a0..63126ba 100644 --- a/docs/source/auto_examples/plot_OT_2D_samples.py +++ b/docs/source/auto_examples/plot_OT_2D_samples.py @@ -10,6 +10,7 @@ sum of diracs. The OT matrix is plotted with the samples. """ # Author: Remi Flamary <remi.flamary@unice.fr> +# Kilian Fatras <kilian.fatras@irisa.fr> # # License: MIT License @@ -100,3 +101,28 @@ pl.legend(loc=0) pl.title('OT matrix Sinkhorn with samples') pl.show() + + +############################################################################## +# Emprirical Sinkhorn +# ---------------- + +#%% sinkhorn + +# reg term +lambd = 1e-3 + +Ges = ot.bregman.empirical_sinkhorn(xs, xt, lambd) + +pl.figure(7) +pl.imshow(Ges, interpolation='nearest') +pl.title('OT matrix empirical sinkhorn') + +pl.figure(8) +ot.plot.plot2D_samples_mat(xs, xt, Ges, color=[.5, .5, 1]) +pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples') +pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples') +pl.legend(loc=0) +pl.title('OT matrix Sinkhorn from samples') + +pl.show() diff --git a/docs/source/auto_examples/plot_OT_2D_samples.rst b/docs/source/auto_examples/plot_OT_2D_samples.rst index 624ae3e..1f1d713 100644 --- a/docs/source/auto_examples/plot_OT_2D_samples.rst +++ b/docs/source/auto_examples/plot_OT_2D_samples.rst @@ -17,6 +17,7 @@ sum of diracs. The OT matrix is plotted with the samples. # Author: Remi Flamary <remi.flamary@unice.fr> + # Kilian Fatras <kilian.fatras@irisa.fr> # # License: MIT License @@ -176,6 +177,8 @@ Compute Sinkhorn + + .. rst-class:: sphx-glr-horizontal @@ -192,7 +195,58 @@ Compute Sinkhorn -**Total running time of the script:** ( 0 minutes 3.027 seconds) +Emprirical Sinkhorn +---------------- + + + +.. code-block:: python + + + #%% sinkhorn + + # reg term + lambd = 1e-3 + + Ges = ot.bregman.empirical_sinkhorn(xs, xt, lambd) + + pl.figure(7) + pl.imshow(Ges, interpolation='nearest') + pl.title('OT matrix empirical sinkhorn') + + pl.figure(8) + ot.plot.plot2D_samples_mat(xs, xt, Ges, color=[.5, .5, 1]) + pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples') + pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples') + pl.legend(loc=0) + pl.title('OT matrix Sinkhorn from samples') + + pl.show() + + + +.. rst-class:: sphx-glr-horizontal + + + * + + .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_013.png + :scale: 47 + + * + + .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_014.png + :scale: 47 + + +.. rst-class:: sphx-glr-script-out + + Out:: + + Warning: numerical errors at iteration 0 + + +**Total running time of the script:** ( 0 minutes 2.616 seconds) diff --git a/docs/source/auto_examples/plot_UOT_1D.ipynb b/docs/source/auto_examples/plot_UOT_1D.ipynb new file mode 100644 index 0000000..c695306 --- /dev/null +++ b/docs/source/auto_examples/plot_UOT_1D.ipynb @@ -0,0 +1,108 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n# 1D Unbalanced optimal transport\n\n\nThis example illustrates the computation of Unbalanced Optimal transport\nusing a Kullback-Leibler relaxation.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Author: Hicham Janati <hicham.janati@inria.fr>\n#\n# License: MIT License\n\nimport numpy as np\nimport matplotlib.pylab as pl\nimport ot\nimport ot.plot\nfrom ot.datasets import make_1D_gauss as gauss" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Generate data\n-------------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#%% parameters\n\nn = 100 # nb bins\n\n# bin positions\nx = np.arange(n, dtype=np.float64)\n\n# Gaussian distributions\na = gauss(n, m=20, s=5) # m= mean, s= std\nb = gauss(n, m=60, s=10)\n\n# make distributions unbalanced\nb *= 5.\n\n# loss matrix\nM = ot.dist(x.reshape((n, 1)), x.reshape((n, 1)))\nM /= M.max()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Plot distributions and loss matrix\n----------------------------------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#%% plot the distributions\n\npl.figure(1, figsize=(6.4, 3))\npl.plot(x, a, 'b', label='Source distribution')\npl.plot(x, b, 'r', label='Target distribution')\npl.legend()\n\n# plot distributions and loss matrix\n\npl.figure(2, figsize=(5, 5))\not.plot.plot1D_mat(a, b, M, 'Cost matrix M')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Solve Unbalanced Sinkhorn\n--------------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Sinkhorn\n\nepsilon = 0.1 # entropy parameter\nalpha = 1. # Unbalanced KL relaxation parameter\nGs = ot.unbalanced.sinkhorn_unbalanced(a, b, M, epsilon, alpha, verbose=True)\n\npl.figure(4, figsize=(5, 5))\not.plot.plot1D_mat(a, b, Gs, 'UOT matrix Sinkhorn')\n\npl.show()" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.6.8" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +}
\ No newline at end of file diff --git a/docs/source/auto_examples/plot_UOT_1D.py b/docs/source/auto_examples/plot_UOT_1D.py new file mode 100644 index 0000000..2ea8b05 --- /dev/null +++ b/docs/source/auto_examples/plot_UOT_1D.py @@ -0,0 +1,76 @@ +# -*- coding: utf-8 -*- +""" +=============================== +1D Unbalanced optimal transport +=============================== + +This example illustrates the computation of Unbalanced Optimal transport +using a Kullback-Leibler relaxation. +""" + +# Author: Hicham Janati <hicham.janati@inria.fr> +# +# License: MIT License + +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 +# ------------- + + +#%% parameters + +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) + +# make distributions unbalanced +b *= 5. + +# loss matrix +M = ot.dist(x.reshape((n, 1)), x.reshape((n, 1))) +M /= M.max() + + +############################################################################## +# Plot distributions and loss matrix +# ---------------------------------- + +#%% plot the distributions + +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 distributions and loss matrix + +pl.figure(2, figsize=(5, 5)) +ot.plot.plot1D_mat(a, b, M, 'Cost matrix M') + + +############################################################################## +# Solve Unbalanced Sinkhorn +# -------------- + + +# Sinkhorn + +epsilon = 0.1 # entropy parameter +alpha = 1. # Unbalanced KL relaxation parameter +Gs = ot.unbalanced.sinkhorn_unbalanced(a, b, M, epsilon, alpha, verbose=True) + +pl.figure(4, figsize=(5, 5)) +ot.plot.plot1D_mat(a, b, Gs, 'UOT matrix Sinkhorn') + +pl.show() diff --git a/docs/source/auto_examples/plot_UOT_1D.rst b/docs/source/auto_examples/plot_UOT_1D.rst new file mode 100644 index 0000000..8e618b4 --- /dev/null +++ b/docs/source/auto_examples/plot_UOT_1D.rst @@ -0,0 +1,173 @@ + + +.. _sphx_glr_auto_examples_plot_UOT_1D.py: + + +=============================== +1D Unbalanced optimal transport +=============================== + +This example illustrates the computation of Unbalanced Optimal transport +using a Kullback-Leibler relaxation. + + + +.. code-block:: python + + + # Author: Hicham Janati <hicham.janati@inria.fr> + # + # License: MIT License + + 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 +------------- + + + +.. code-block:: python + + + + #%% parameters + + 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) + + # make distributions unbalanced + b *= 5. + + # loss matrix + M = ot.dist(x.reshape((n, 1)), x.reshape((n, 1))) + M /= M.max() + + + + + + + + +Plot distributions and loss matrix +---------------------------------- + + + +.. code-block:: python + + + #%% plot the distributions + + 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 distributions and loss matrix + + pl.figure(2, figsize=(5, 5)) + ot.plot.plot1D_mat(a, b, M, 'Cost matrix M') + + + + + +.. rst-class:: sphx-glr-horizontal + + + * + + .. image:: /auto_examples/images/sphx_glr_plot_UOT_1D_001.png + :scale: 47 + + * + + .. image:: /auto_examples/images/sphx_glr_plot_UOT_1D_002.png + :scale: 47 + + + + +Solve Unbalanced Sinkhorn +-------------- + + + +.. code-block:: python + + + + # Sinkhorn + + epsilon = 0.1 # entropy parameter + alpha = 1. # Unbalanced KL relaxation parameter + Gs = ot.unbalanced.sinkhorn_unbalanced(a, b, M, epsilon, alpha, verbose=True) + + pl.figure(4, figsize=(5, 5)) + ot.plot.plot1D_mat(a, b, Gs, 'UOT matrix Sinkhorn') + + pl.show() + + + +.. image:: /auto_examples/images/sphx_glr_plot_UOT_1D_006.png + :align: center + + +.. rst-class:: sphx-glr-script-out + + Out:: + + It. |Err + ------------------- + 0|1.838786e+00| + 10|1.242379e-01| + 20|2.581314e-03| + 30|5.674552e-05| + 40|1.252959e-06| + 50|2.768136e-08| + 60|6.116090e-10| + + +**Total running time of the script:** ( 0 minutes 0.259 seconds) + + + +.. only :: html + + .. container:: sphx-glr-footer + + + .. container:: sphx-glr-download + + :download:`Download Python source code: plot_UOT_1D.py <plot_UOT_1D.py>` + + + + .. container:: sphx-glr-download + + :download:`Download Jupyter notebook: plot_UOT_1D.ipynb <plot_UOT_1D.ipynb>` + + +.. only:: html + + .. rst-class:: sphx-glr-signature + + `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_ diff --git a/docs/source/auto_examples/plot_UOT_barycenter_1D.ipynb b/docs/source/auto_examples/plot_UOT_barycenter_1D.ipynb new file mode 100644 index 0000000..e59cdc2 --- /dev/null +++ b/docs/source/auto_examples/plot_UOT_barycenter_1D.ipynb @@ -0,0 +1,126 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n# 1D Wasserstein barycenter demo for Unbalanced distributions\n\n\nThis example illustrates the computation of regularized Wassersyein Barycenter\nas proposed in [10] for Unbalanced inputs.\n\n\n[10] Chizat, L., Peyr\u00e9, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.\n\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Author: Hicham Janati <hicham.janati@inria.fr>\n#\n# License: MIT License\n\nimport numpy as np\nimport matplotlib.pylab as pl\nimport ot\n# necessary for 3d plot even if not used\nfrom mpl_toolkits.mplot3d import Axes3D # noqa\nfrom matplotlib.collections import PolyCollection" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Generate data\n-------------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# parameters\n\nn = 100 # nb bins\n\n# bin positions\nx = np.arange(n, dtype=np.float64)\n\n# Gaussian distributions\na1 = ot.datasets.make_1D_gauss(n, m=20, s=5) # m= mean, s= std\na2 = ot.datasets.make_1D_gauss(n, m=60, s=8)\n\n# make unbalanced dists\na2 *= 3.\n\n# creating matrix A containing all distributions\nA = np.vstack((a1, a2)).T\nn_distributions = A.shape[1]\n\n# loss matrix + normalization\nM = ot.utils.dist0(n)\nM /= M.max()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Plot data\n---------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# plot the distributions\n\npl.figure(1, figsize=(6.4, 3))\nfor i in range(n_distributions):\n pl.plot(x, A[:, i])\npl.title('Distributions')\npl.tight_layout()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Barycenter computation\n----------------------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# non weighted barycenter computation\n\nweight = 0.5 # 0<=weight<=1\nweights = np.array([1 - weight, weight])\n\n# l2bary\nbary_l2 = A.dot(weights)\n\n# wasserstein\nreg = 1e-3\nalpha = 1.\n\nbary_wass = ot.unbalanced.barycenter_unbalanced(A, M, reg, alpha, weights)\n\npl.figure(2)\npl.clf()\npl.subplot(2, 1, 1)\nfor i in range(n_distributions):\n pl.plot(x, A[:, i])\npl.title('Distributions')\n\npl.subplot(2, 1, 2)\npl.plot(x, bary_l2, 'r', label='l2')\npl.plot(x, bary_wass, 'g', label='Wasserstein')\npl.legend()\npl.title('Barycenters')\npl.tight_layout()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Barycentric interpolation\n-------------------------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# barycenter interpolation\n\nn_weight = 11\nweight_list = np.linspace(0, 1, n_weight)\n\n\nB_l2 = np.zeros((n, n_weight))\n\nB_wass = np.copy(B_l2)\n\nfor i in range(0, n_weight):\n weight = weight_list[i]\n weights = np.array([1 - weight, weight])\n B_l2[:, i] = A.dot(weights)\n B_wass[:, i] = ot.unbalanced.barycenter_unbalanced(A, M, reg, alpha, weights)\n\n\n# plot interpolation\n\npl.figure(3)\n\ncmap = pl.cm.get_cmap('viridis')\nverts = []\nzs = weight_list\nfor i, z in enumerate(zs):\n ys = B_l2[:, i]\n verts.append(list(zip(x, ys)))\n\nax = pl.gcf().gca(projection='3d')\n\npoly = PolyCollection(verts, facecolors=[cmap(a) for a in weight_list])\npoly.set_alpha(0.7)\nax.add_collection3d(poly, zs=zs, zdir='y')\nax.set_xlabel('x')\nax.set_xlim3d(0, n)\nax.set_ylabel(r'$\\alpha$')\nax.set_ylim3d(0, 1)\nax.set_zlabel('')\nax.set_zlim3d(0, B_l2.max() * 1.01)\npl.title('Barycenter interpolation with l2')\npl.tight_layout()\n\npl.figure(4)\ncmap = pl.cm.get_cmap('viridis')\nverts = []\nzs = weight_list\nfor i, z in enumerate(zs):\n ys = B_wass[:, i]\n verts.append(list(zip(x, ys)))\n\nax = pl.gcf().gca(projection='3d')\n\npoly = PolyCollection(verts, facecolors=[cmap(a) for a in weight_list])\npoly.set_alpha(0.7)\nax.add_collection3d(poly, zs=zs, zdir='y')\nax.set_xlabel('x')\nax.set_xlim3d(0, n)\nax.set_ylabel(r'$\\alpha$')\nax.set_ylim3d(0, 1)\nax.set_zlabel('')\nax.set_zlim3d(0, B_l2.max() * 1.01)\npl.title('Barycenter interpolation with Wasserstein')\npl.tight_layout()\n\npl.show()" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.6.8" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +}
\ No newline at end of file diff --git a/docs/source/auto_examples/plot_UOT_barycenter_1D.py b/docs/source/auto_examples/plot_UOT_barycenter_1D.py new file mode 100644 index 0000000..c8d9d3b --- /dev/null +++ b/docs/source/auto_examples/plot_UOT_barycenter_1D.py @@ -0,0 +1,164 @@ +# -*- coding: utf-8 -*- +""" +=========================================================== +1D Wasserstein barycenter demo for Unbalanced distributions +=========================================================== + +This example illustrates the computation of regularized Wassersyein Barycenter +as proposed in [10] for Unbalanced inputs. + + +[10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816. + +""" + +# Author: Hicham Janati <hicham.janati@inria.fr> +# +# License: MIT License + +import numpy as np +import matplotlib.pylab as pl +import ot +# necessary for 3d plot even if not used +from mpl_toolkits.mplot3d import Axes3D # noqa +from matplotlib.collections import PolyCollection + +############################################################################## +# Generate data +# ------------- + +# parameters + +n = 100 # nb bins + +# bin positions +x = np.arange(n, dtype=np.float64) + +# Gaussian distributions +a1 = ot.datasets.make_1D_gauss(n, m=20, s=5) # m= mean, s= std +a2 = ot.datasets.make_1D_gauss(n, m=60, s=8) + +# make unbalanced dists +a2 *= 3. + +# creating matrix A containing all distributions +A = np.vstack((a1, a2)).T +n_distributions = A.shape[1] + +# loss matrix + normalization +M = ot.utils.dist0(n) +M /= M.max() + +############################################################################## +# Plot data +# --------- + +# plot the distributions + +pl.figure(1, figsize=(6.4, 3)) +for i in range(n_distributions): + pl.plot(x, A[:, i]) +pl.title('Distributions') +pl.tight_layout() + +############################################################################## +# Barycenter computation +# ---------------------- + +# non weighted barycenter computation + +weight = 0.5 # 0<=weight<=1 +weights = np.array([1 - weight, weight]) + +# l2bary +bary_l2 = A.dot(weights) + +# wasserstein +reg = 1e-3 +alpha = 1. + +bary_wass = ot.unbalanced.barycenter_unbalanced(A, M, reg, alpha, weights) + +pl.figure(2) +pl.clf() +pl.subplot(2, 1, 1) +for i in range(n_distributions): + pl.plot(x, A[:, i]) +pl.title('Distributions') + +pl.subplot(2, 1, 2) +pl.plot(x, bary_l2, 'r', label='l2') +pl.plot(x, bary_wass, 'g', label='Wasserstein') +pl.legend() +pl.title('Barycenters') +pl.tight_layout() + +############################################################################## +# Barycentric interpolation +# ------------------------- + +# barycenter interpolation + +n_weight = 11 +weight_list = np.linspace(0, 1, n_weight) + + +B_l2 = np.zeros((n, n_weight)) + +B_wass = np.copy(B_l2) + +for i in range(0, n_weight): + weight = weight_list[i] + weights = np.array([1 - weight, weight]) + B_l2[:, i] = A.dot(weights) + B_wass[:, i] = ot.unbalanced.barycenter_unbalanced(A, M, reg, alpha, weights) + + +# plot interpolation + +pl.figure(3) + +cmap = pl.cm.get_cmap('viridis') +verts = [] +zs = weight_list +for i, z in enumerate(zs): + ys = B_l2[:, i] + verts.append(list(zip(x, ys))) + +ax = pl.gcf().gca(projection='3d') + +poly = PolyCollection(verts, facecolors=[cmap(a) for a in weight_list]) +poly.set_alpha(0.7) +ax.add_collection3d(poly, zs=zs, zdir='y') +ax.set_xlabel('x') +ax.set_xlim3d(0, n) +ax.set_ylabel(r'$\alpha$') +ax.set_ylim3d(0, 1) +ax.set_zlabel('') +ax.set_zlim3d(0, B_l2.max() * 1.01) +pl.title('Barycenter interpolation with l2') +pl.tight_layout() + +pl.figure(4) +cmap = pl.cm.get_cmap('viridis') +verts = [] +zs = weight_list +for i, z in enumerate(zs): + ys = B_wass[:, i] + verts.append(list(zip(x, ys))) + +ax = pl.gcf().gca(projection='3d') + +poly = PolyCollection(verts, facecolors=[cmap(a) for a in weight_list]) +poly.set_alpha(0.7) +ax.add_collection3d(poly, zs=zs, zdir='y') +ax.set_xlabel('x') +ax.set_xlim3d(0, n) +ax.set_ylabel(r'$\alpha$') +ax.set_ylim3d(0, 1) +ax.set_zlabel('') +ax.set_zlim3d(0, B_l2.max() * 1.01) +pl.title('Barycenter interpolation with Wasserstein') +pl.tight_layout() + +pl.show() diff --git a/docs/source/auto_examples/plot_UOT_barycenter_1D.rst b/docs/source/auto_examples/plot_UOT_barycenter_1D.rst new file mode 100644 index 0000000..ac17587 --- /dev/null +++ b/docs/source/auto_examples/plot_UOT_barycenter_1D.rst @@ -0,0 +1,261 @@ + + +.. _sphx_glr_auto_examples_plot_UOT_barycenter_1D.py: + + +=========================================================== +1D Wasserstein barycenter demo for Unbalanced distributions +=========================================================== + +This example illustrates the computation of regularized Wassersyein Barycenter +as proposed in [10] for Unbalanced inputs. + + +[10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816. + + + + +.. code-block:: python + + + # Author: Hicham Janati <hicham.janati@inria.fr> + # + # License: MIT License + + import numpy as np + import matplotlib.pylab as pl + import ot + # necessary for 3d plot even if not used + from mpl_toolkits.mplot3d import Axes3D # noqa + from matplotlib.collections import PolyCollection + + + + + + + +Generate data +------------- + + + +.. code-block:: python + + + # parameters + + n = 100 # nb bins + + # bin positions + x = np.arange(n, dtype=np.float64) + + # Gaussian distributions + a1 = ot.datasets.make_1D_gauss(n, m=20, s=5) # m= mean, s= std + a2 = ot.datasets.make_1D_gauss(n, m=60, s=8) + + # make unbalanced dists + a2 *= 3. + + # creating matrix A containing all distributions + A = np.vstack((a1, a2)).T + n_distributions = A.shape[1] + + # loss matrix + normalization + M = ot.utils.dist0(n) + M /= M.max() + + + + + + + +Plot data +--------- + + + +.. code-block:: python + + + # plot the distributions + + pl.figure(1, figsize=(6.4, 3)) + for i in range(n_distributions): + pl.plot(x, A[:, i]) + pl.title('Distributions') + pl.tight_layout() + + + + +.. image:: /auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_001.png + :align: center + + + + +Barycenter computation +---------------------- + + + +.. code-block:: python + + + # non weighted barycenter computation + + weight = 0.5 # 0<=weight<=1 + weights = np.array([1 - weight, weight]) + + # l2bary + bary_l2 = A.dot(weights) + + # wasserstein + reg = 1e-3 + alpha = 1. + + bary_wass = ot.unbalanced.barycenter_unbalanced(A, M, reg, alpha, weights) + + pl.figure(2) + pl.clf() + pl.subplot(2, 1, 1) + for i in range(n_distributions): + pl.plot(x, A[:, i]) + pl.title('Distributions') + + pl.subplot(2, 1, 2) + pl.plot(x, bary_l2, 'r', label='l2') + pl.plot(x, bary_wass, 'g', label='Wasserstein') + pl.legend() + pl.title('Barycenters') + pl.tight_layout() + + + + +.. image:: /auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_003.png + :align: center + + + + +Barycentric interpolation +------------------------- + + + +.. code-block:: python + + + # barycenter interpolation + + n_weight = 11 + weight_list = np.linspace(0, 1, n_weight) + + + B_l2 = np.zeros((n, n_weight)) + + B_wass = np.copy(B_l2) + + for i in range(0, n_weight): + weight = weight_list[i] + weights = np.array([1 - weight, weight]) + B_l2[:, i] = A.dot(weights) + B_wass[:, i] = ot.unbalanced.barycenter_unbalanced(A, M, reg, alpha, weights) + + + # plot interpolation + + pl.figure(3) + + cmap = pl.cm.get_cmap('viridis') + verts = [] + zs = weight_list + for i, z in enumerate(zs): + ys = B_l2[:, i] + verts.append(list(zip(x, ys))) + + ax = pl.gcf().gca(projection='3d') + + poly = PolyCollection(verts, facecolors=[cmap(a) for a in weight_list]) + poly.set_alpha(0.7) + ax.add_collection3d(poly, zs=zs, zdir='y') + ax.set_xlabel('x') + ax.set_xlim3d(0, n) + ax.set_ylabel(r'$\alpha$') + ax.set_ylim3d(0, 1) + ax.set_zlabel('') + ax.set_zlim3d(0, B_l2.max() * 1.01) + pl.title('Barycenter interpolation with l2') + pl.tight_layout() + + pl.figure(4) + cmap = pl.cm.get_cmap('viridis') + verts = [] + zs = weight_list + for i, z in enumerate(zs): + ys = B_wass[:, i] + verts.append(list(zip(x, ys))) + + ax = pl.gcf().gca(projection='3d') + + poly = PolyCollection(verts, facecolors=[cmap(a) for a in weight_list]) + poly.set_alpha(0.7) + ax.add_collection3d(poly, zs=zs, zdir='y') + ax.set_xlabel('x') + ax.set_xlim3d(0, n) + ax.set_ylabel(r'$\alpha$') + ax.set_ylim3d(0, 1) + ax.set_zlabel('') + ax.set_zlim3d(0, B_l2.max() * 1.01) + pl.title('Barycenter interpolation with Wasserstein') + pl.tight_layout() + + pl.show() + + + +.. rst-class:: sphx-glr-horizontal + + + * + + .. image:: /auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_005.png + :scale: 47 + + * + + .. image:: /auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_006.png + :scale: 47 + + + + +**Total running time of the script:** ( 0 minutes 0.344 seconds) + + + +.. only :: html + + .. container:: sphx-glr-footer + + + .. container:: sphx-glr-download + + :download:`Download Python source code: plot_UOT_barycenter_1D.py <plot_UOT_barycenter_1D.py>` + + + + .. container:: sphx-glr-download + + :download:`Download Jupyter notebook: plot_UOT_barycenter_1D.ipynb <plot_UOT_barycenter_1D.ipynb>` + + +.. only:: html + + .. rst-class:: sphx-glr-signature + + `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_ diff --git a/docs/source/auto_examples/plot_barycenter_fgw.ipynb b/docs/source/auto_examples/plot_barycenter_fgw.ipynb new file mode 100644 index 0000000..28229b2 --- /dev/null +++ b/docs/source/auto_examples/plot_barycenter_fgw.ipynb @@ -0,0 +1,126 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n=================================\nPlot graphs' barycenter using FGW\n=================================\n\nThis example illustrates the computation barycenter of labeled graphs using FGW\n\nRequires networkx >=2\n\n.. [18] Vayer Titouan, Chapel Laetitia, Flamary R{'e}mi, Tavenard Romain\n and Courty Nicolas\n \"Optimal Transport for structured data with application on graphs\"\n International Conference on Machine Learning (ICML). 2019.\n\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Author: Titouan Vayer <titouan.vayer@irisa.fr>\n#\n# License: MIT License\n\n#%% load libraries\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport networkx as nx\nimport math\nfrom scipy.sparse.csgraph import shortest_path\nimport matplotlib.colors as mcol\nfrom matplotlib import cm\nfrom ot.gromov import fgw_barycenters\n#%% Graph functions\n\n\ndef find_thresh(C, inf=0.5, sup=3, step=10):\n \"\"\" Trick to find the adequate thresholds from where value of the C matrix are considered close enough to say that nodes are connected\n Tthe threshold is found by a linesearch between values \"inf\" and \"sup\" with \"step\" thresholds tested.\n The optimal threshold is the one which minimizes the reconstruction error between the shortest_path matrix coming from the thresholded adjency matrix\n and the original matrix.\n Parameters\n ----------\n C : ndarray, shape (n_nodes,n_nodes)\n The structure matrix to threshold\n inf : float\n The beginning of the linesearch\n sup : float\n The end of the linesearch\n step : integer\n Number of thresholds tested\n \"\"\"\n dist = []\n search = np.linspace(inf, sup, step)\n for thresh in search:\n Cprime = sp_to_adjency(C, 0, thresh)\n SC = shortest_path(Cprime, method='D')\n SC[SC == float('inf')] = 100\n dist.append(np.linalg.norm(SC - C))\n return search[np.argmin(dist)], dist\n\n\ndef sp_to_adjency(C, threshinf=0.2, threshsup=1.8):\n \"\"\" Thresholds the structure matrix in order to compute an adjency matrix.\n All values between threshinf and threshsup are considered representing connected nodes and set to 1. Else are set to 0\n Parameters\n ----------\n C : ndarray, shape (n_nodes,n_nodes)\n The structure matrix to threshold\n threshinf : float\n The minimum value of distance from which the new value is set to 1\n threshsup : float\n The maximum value of distance from which the new value is set to 1\n Returns\n -------\n C : ndarray, shape (n_nodes,n_nodes)\n The threshold matrix. Each element is in {0,1}\n \"\"\"\n H = np.zeros_like(C)\n np.fill_diagonal(H, np.diagonal(C))\n C = C - H\n C = np.minimum(np.maximum(C, threshinf), threshsup)\n C[C == threshsup] = 0\n C[C != 0] = 1\n\n return C\n\n\ndef build_noisy_circular_graph(N=20, mu=0, sigma=0.3, with_noise=False, structure_noise=False, p=None):\n \"\"\" Create a noisy circular graph\n \"\"\"\n g = nx.Graph()\n g.add_nodes_from(list(range(N)))\n for i in range(N):\n noise = float(np.random.normal(mu, sigma, 1))\n if with_noise:\n g.add_node(i, attr_name=math.sin((2 * i * math.pi / N)) + noise)\n else:\n g.add_node(i, attr_name=math.sin(2 * i * math.pi / N))\n g.add_edge(i, i + 1)\n if structure_noise:\n randomint = np.random.randint(0, p)\n if randomint == 0:\n if i <= N - 3:\n g.add_edge(i, i + 2)\n if i == N - 2:\n g.add_edge(i, 0)\n if i == N - 1:\n g.add_edge(i, 1)\n g.add_edge(N, 0)\n noise = float(np.random.normal(mu, sigma, 1))\n if with_noise:\n g.add_node(N, attr_name=math.sin((2 * N * math.pi / N)) + noise)\n else:\n g.add_node(N, attr_name=math.sin(2 * N * math.pi / N))\n return g\n\n\ndef graph_colors(nx_graph, vmin=0, vmax=7):\n cnorm = mcol.Normalize(vmin=vmin, vmax=vmax)\n cpick = cm.ScalarMappable(norm=cnorm, cmap='viridis')\n cpick.set_array([])\n val_map = {}\n for k, v in nx.get_node_attributes(nx_graph, 'attr_name').items():\n val_map[k] = cpick.to_rgba(v)\n colors = []\n for node in nx_graph.nodes():\n colors.append(val_map[node])\n return colors" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Generate data\n-------------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#%% circular dataset\n# We build a dataset of noisy circular graphs.\n# Noise is added on the structures by random connections and on the features by gaussian noise.\n\n\nnp.random.seed(30)\nX0 = []\nfor k in range(9):\n X0.append(build_noisy_circular_graph(np.random.randint(15, 25), with_noise=True, structure_noise=True, p=3))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Plot data\n---------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#%% Plot graphs\n\nplt.figure(figsize=(8, 10))\nfor i in range(len(X0)):\n plt.subplot(3, 3, i + 1)\n g = X0[i]\n pos = nx.kamada_kawai_layout(g)\n nx.draw(g, pos=pos, node_color=graph_colors(g, vmin=-1, vmax=1), with_labels=False, node_size=100)\nplt.suptitle('Dataset of noisy graphs. Color indicates the label', fontsize=20)\nplt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Barycenter computation\n----------------------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#%% We compute the barycenter using FGW. Structure matrices are computed using the shortest_path distance in the graph\n# Features distances are the euclidean distances\nCs = [shortest_path(nx.adjacency_matrix(x)) for x in X0]\nps = [np.ones(len(x.nodes())) / len(x.nodes()) for x in X0]\nYs = [np.array([v for (k, v) in nx.get_node_attributes(x, 'attr_name').items()]).reshape(-1, 1) for x in X0]\nlambdas = np.array([np.ones(len(Ys)) / len(Ys)]).ravel()\nsizebary = 15 # we choose a barycenter with 15 nodes\n\nA, C, log = fgw_barycenters(sizebary, Ys, Cs, ps, lambdas, alpha=0.95, log=True)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Plot Barycenter\n-------------------------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#%% Create the barycenter\nbary = nx.from_numpy_matrix(sp_to_adjency(C, threshinf=0, threshsup=find_thresh(C, sup=100, step=100)[0]))\nfor i, v in enumerate(A.ravel()):\n bary.add_node(i, attr_name=v)\n\n#%%\npos = nx.kamada_kawai_layout(bary)\nnx.draw(bary, pos=pos, node_color=graph_colors(bary, vmin=-1, vmax=1), with_labels=False)\nplt.suptitle('Barycenter', fontsize=20)\nplt.show()" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.6.8" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +}
\ No newline at end of file diff --git a/docs/source/auto_examples/plot_barycenter_fgw.py b/docs/source/auto_examples/plot_barycenter_fgw.py new file mode 100644 index 0000000..77b0370 --- /dev/null +++ b/docs/source/auto_examples/plot_barycenter_fgw.py @@ -0,0 +1,184 @@ +# -*- coding: utf-8 -*- +""" +================================= +Plot graphs' barycenter using FGW +================================= + +This example illustrates the computation barycenter of labeled graphs using FGW + +Requires networkx >=2 + +.. [18] Vayer Titouan, Chapel Laetitia, Flamary R{\'e}mi, Tavenard Romain + and Courty Nicolas + "Optimal Transport for structured data with application on graphs" + International Conference on Machine Learning (ICML). 2019. + +""" + +# Author: Titouan Vayer <titouan.vayer@irisa.fr> +# +# License: MIT License + +#%% load libraries +import numpy as np +import matplotlib.pyplot as plt +import networkx as nx +import math +from scipy.sparse.csgraph import shortest_path +import matplotlib.colors as mcol +from matplotlib import cm +from ot.gromov import fgw_barycenters +#%% Graph functions + + +def find_thresh(C, inf=0.5, sup=3, step=10): + """ Trick to find the adequate thresholds from where value of the C matrix are considered close enough to say that nodes are connected + Tthe threshold is found by a linesearch between values "inf" and "sup" with "step" thresholds tested. + The optimal threshold is the one which minimizes the reconstruction error between the shortest_path matrix coming from the thresholded adjency matrix + and the original matrix. + Parameters + ---------- + C : ndarray, shape (n_nodes,n_nodes) + The structure matrix to threshold + inf : float + The beginning of the linesearch + sup : float + The end of the linesearch + step : integer + Number of thresholds tested + """ + dist = [] + search = np.linspace(inf, sup, step) + for thresh in search: + Cprime = sp_to_adjency(C, 0, thresh) + SC = shortest_path(Cprime, method='D') + SC[SC == float('inf')] = 100 + dist.append(np.linalg.norm(SC - C)) + return search[np.argmin(dist)], dist + + +def sp_to_adjency(C, threshinf=0.2, threshsup=1.8): + """ Thresholds the structure matrix in order to compute an adjency matrix. + All values between threshinf and threshsup are considered representing connected nodes and set to 1. Else are set to 0 + Parameters + ---------- + C : ndarray, shape (n_nodes,n_nodes) + The structure matrix to threshold + threshinf : float + The minimum value of distance from which the new value is set to 1 + threshsup : float + The maximum value of distance from which the new value is set to 1 + Returns + ------- + C : ndarray, shape (n_nodes,n_nodes) + The threshold matrix. Each element is in {0,1} + """ + H = np.zeros_like(C) + np.fill_diagonal(H, np.diagonal(C)) + C = C - H + C = np.minimum(np.maximum(C, threshinf), threshsup) + C[C == threshsup] = 0 + C[C != 0] = 1 + + return C + + +def build_noisy_circular_graph(N=20, mu=0, sigma=0.3, with_noise=False, structure_noise=False, p=None): + """ Create a noisy circular graph + """ + g = nx.Graph() + g.add_nodes_from(list(range(N))) + for i in range(N): + noise = float(np.random.normal(mu, sigma, 1)) + if with_noise: + g.add_node(i, attr_name=math.sin((2 * i * math.pi / N)) + noise) + else: + g.add_node(i, attr_name=math.sin(2 * i * math.pi / N)) + g.add_edge(i, i + 1) + if structure_noise: + randomint = np.random.randint(0, p) + if randomint == 0: + if i <= N - 3: + g.add_edge(i, i + 2) + if i == N - 2: + g.add_edge(i, 0) + if i == N - 1: + g.add_edge(i, 1) + g.add_edge(N, 0) + noise = float(np.random.normal(mu, sigma, 1)) + if with_noise: + g.add_node(N, attr_name=math.sin((2 * N * math.pi / N)) + noise) + else: + g.add_node(N, attr_name=math.sin(2 * N * math.pi / N)) + return g + + +def graph_colors(nx_graph, vmin=0, vmax=7): + cnorm = mcol.Normalize(vmin=vmin, vmax=vmax) + cpick = cm.ScalarMappable(norm=cnorm, cmap='viridis') + cpick.set_array([]) + val_map = {} + for k, v in nx.get_node_attributes(nx_graph, 'attr_name').items(): + val_map[k] = cpick.to_rgba(v) + colors = [] + for node in nx_graph.nodes(): + colors.append(val_map[node]) + return colors + +############################################################################## +# Generate data +# ------------- + +#%% circular dataset +# We build a dataset of noisy circular graphs. +# Noise is added on the structures by random connections and on the features by gaussian noise. + + +np.random.seed(30) +X0 = [] +for k in range(9): + X0.append(build_noisy_circular_graph(np.random.randint(15, 25), with_noise=True, structure_noise=True, p=3)) + +############################################################################## +# Plot data +# --------- + +#%% Plot graphs + +plt.figure(figsize=(8, 10)) +for i in range(len(X0)): + plt.subplot(3, 3, i + 1) + g = X0[i] + pos = nx.kamada_kawai_layout(g) + nx.draw(g, pos=pos, node_color=graph_colors(g, vmin=-1, vmax=1), with_labels=False, node_size=100) +plt.suptitle('Dataset of noisy graphs. Color indicates the label', fontsize=20) +plt.show() + +############################################################################## +# Barycenter computation +# ---------------------- + +#%% We compute the barycenter using FGW. Structure matrices are computed using the shortest_path distance in the graph +# Features distances are the euclidean distances +Cs = [shortest_path(nx.adjacency_matrix(x)) for x in X0] +ps = [np.ones(len(x.nodes())) / len(x.nodes()) for x in X0] +Ys = [np.array([v for (k, v) in nx.get_node_attributes(x, 'attr_name').items()]).reshape(-1, 1) for x in X0] +lambdas = np.array([np.ones(len(Ys)) / len(Ys)]).ravel() +sizebary = 15 # we choose a barycenter with 15 nodes + +A, C, log = fgw_barycenters(sizebary, Ys, Cs, ps, lambdas, alpha=0.95, log=True) + +############################################################################## +# Plot Barycenter +# ------------------------- + +#%% Create the barycenter +bary = nx.from_numpy_matrix(sp_to_adjency(C, threshinf=0, threshsup=find_thresh(C, sup=100, step=100)[0])) +for i, v in enumerate(A.ravel()): + bary.add_node(i, attr_name=v) + +#%% +pos = nx.kamada_kawai_layout(bary) +nx.draw(bary, pos=pos, node_color=graph_colors(bary, vmin=-1, vmax=1), with_labels=False) +plt.suptitle('Barycenter', fontsize=20) +plt.show() diff --git a/docs/source/auto_examples/plot_barycenter_fgw.rst b/docs/source/auto_examples/plot_barycenter_fgw.rst new file mode 100644 index 0000000..2c44a65 --- /dev/null +++ b/docs/source/auto_examples/plot_barycenter_fgw.rst @@ -0,0 +1,268 @@ + + +.. _sphx_glr_auto_examples_plot_barycenter_fgw.py: + + +================================= +Plot graphs' barycenter using FGW +================================= + +This example illustrates the computation barycenter of labeled graphs using FGW + +Requires networkx >=2 + +.. [18] Vayer Titouan, Chapel Laetitia, Flamary R{'e}mi, Tavenard Romain + and Courty Nicolas + "Optimal Transport for structured data with application on graphs" + International Conference on Machine Learning (ICML). 2019. + + + + +.. code-block:: python + + + # Author: Titouan Vayer <titouan.vayer@irisa.fr> + # + # License: MIT License + + #%% load libraries + import numpy as np + import matplotlib.pyplot as plt + import networkx as nx + import math + from scipy.sparse.csgraph import shortest_path + import matplotlib.colors as mcol + from matplotlib import cm + from ot.gromov import fgw_barycenters + #%% Graph functions + + + def find_thresh(C, inf=0.5, sup=3, step=10): + """ Trick to find the adequate thresholds from where value of the C matrix are considered close enough to say that nodes are connected + Tthe threshold is found by a linesearch between values "inf" and "sup" with "step" thresholds tested. + The optimal threshold is the one which minimizes the reconstruction error between the shortest_path matrix coming from the thresholded adjency matrix + and the original matrix. + Parameters + ---------- + C : ndarray, shape (n_nodes,n_nodes) + The structure matrix to threshold + inf : float + The beginning of the linesearch + sup : float + The end of the linesearch + step : integer + Number of thresholds tested + """ + dist = [] + search = np.linspace(inf, sup, step) + for thresh in search: + Cprime = sp_to_adjency(C, 0, thresh) + SC = shortest_path(Cprime, method='D') + SC[SC == float('inf')] = 100 + dist.append(np.linalg.norm(SC - C)) + return search[np.argmin(dist)], dist + + + def sp_to_adjency(C, threshinf=0.2, threshsup=1.8): + """ Thresholds the structure matrix in order to compute an adjency matrix. + All values between threshinf and threshsup are considered representing connected nodes and set to 1. Else are set to 0 + Parameters + ---------- + C : ndarray, shape (n_nodes,n_nodes) + The structure matrix to threshold + threshinf : float + The minimum value of distance from which the new value is set to 1 + threshsup : float + The maximum value of distance from which the new value is set to 1 + Returns + ------- + C : ndarray, shape (n_nodes,n_nodes) + The threshold matrix. Each element is in {0,1} + """ + H = np.zeros_like(C) + np.fill_diagonal(H, np.diagonal(C)) + C = C - H + C = np.minimum(np.maximum(C, threshinf), threshsup) + C[C == threshsup] = 0 + C[C != 0] = 1 + + return C + + + def build_noisy_circular_graph(N=20, mu=0, sigma=0.3, with_noise=False, structure_noise=False, p=None): + """ Create a noisy circular graph + """ + g = nx.Graph() + g.add_nodes_from(list(range(N))) + for i in range(N): + noise = float(np.random.normal(mu, sigma, 1)) + if with_noise: + g.add_node(i, attr_name=math.sin((2 * i * math.pi / N)) + noise) + else: + g.add_node(i, attr_name=math.sin(2 * i * math.pi / N)) + g.add_edge(i, i + 1) + if structure_noise: + randomint = np.random.randint(0, p) + if randomint == 0: + if i <= N - 3: + g.add_edge(i, i + 2) + if i == N - 2: + g.add_edge(i, 0) + if i == N - 1: + g.add_edge(i, 1) + g.add_edge(N, 0) + noise = float(np.random.normal(mu, sigma, 1)) + if with_noise: + g.add_node(N, attr_name=math.sin((2 * N * math.pi / N)) + noise) + else: + g.add_node(N, attr_name=math.sin(2 * N * math.pi / N)) + return g + + + def graph_colors(nx_graph, vmin=0, vmax=7): + cnorm = mcol.Normalize(vmin=vmin, vmax=vmax) + cpick = cm.ScalarMappable(norm=cnorm, cmap='viridis') + cpick.set_array([]) + val_map = {} + for k, v in nx.get_node_attributes(nx_graph, 'attr_name').items(): + val_map[k] = cpick.to_rgba(v) + colors = [] + for node in nx_graph.nodes(): + colors.append(val_map[node]) + return colors + + + + + + + +Generate data +------------- + + + +.. code-block:: python + + + #%% circular dataset + # We build a dataset of noisy circular graphs. + # Noise is added on the structures by random connections and on the features by gaussian noise. + + + np.random.seed(30) + X0 = [] + for k in range(9): + X0.append(build_noisy_circular_graph(np.random.randint(15, 25), with_noise=True, structure_noise=True, p=3)) + + + + + + + +Plot data +--------- + + + +.. code-block:: python + + + #%% Plot graphs + + plt.figure(figsize=(8, 10)) + for i in range(len(X0)): + plt.subplot(3, 3, i + 1) + g = X0[i] + pos = nx.kamada_kawai_layout(g) + nx.draw(g, pos=pos, node_color=graph_colors(g, vmin=-1, vmax=1), with_labels=False, node_size=100) + plt.suptitle('Dataset of noisy graphs. Color indicates the label', fontsize=20) + plt.show() + + + + +.. image:: /auto_examples/images/sphx_glr_plot_barycenter_fgw_001.png + :align: center + + + + +Barycenter computation +---------------------- + + + +.. code-block:: python + + + #%% We compute the barycenter using FGW. Structure matrices are computed using the shortest_path distance in the graph + # Features distances are the euclidean distances + Cs = [shortest_path(nx.adjacency_matrix(x)) for x in X0] + ps = [np.ones(len(x.nodes())) / len(x.nodes()) for x in X0] + Ys = [np.array([v for (k, v) in nx.get_node_attributes(x, 'attr_name').items()]).reshape(-1, 1) for x in X0] + lambdas = np.array([np.ones(len(Ys)) / len(Ys)]).ravel() + sizebary = 15 # we choose a barycenter with 15 nodes + + A, C, log = fgw_barycenters(sizebary, Ys, Cs, ps, lambdas, alpha=0.95, log=True) + + + + + + + +Plot Barycenter +------------------------- + + + +.. code-block:: python + + + #%% Create the barycenter + bary = nx.from_numpy_matrix(sp_to_adjency(C, threshinf=0, threshsup=find_thresh(C, sup=100, step=100)[0])) + for i, v in enumerate(A.ravel()): + bary.add_node(i, attr_name=v) + + #%% + pos = nx.kamada_kawai_layout(bary) + nx.draw(bary, pos=pos, node_color=graph_colors(bary, vmin=-1, vmax=1), with_labels=False) + plt.suptitle('Barycenter', fontsize=20) + plt.show() + + + +.. image:: /auto_examples/images/sphx_glr_plot_barycenter_fgw_002.png + :align: center + + + + +**Total running time of the script:** ( 0 minutes 2.065 seconds) + + + +.. only :: html + + .. container:: sphx-glr-footer + + + .. container:: sphx-glr-download + + :download:`Download Python source code: plot_barycenter_fgw.py <plot_barycenter_fgw.py>` + + + + .. container:: sphx-glr-download + + :download:`Download Jupyter notebook: plot_barycenter_fgw.ipynb <plot_barycenter_fgw.ipynb>` + + +.. only:: html + + .. rst-class:: sphx-glr-signature + + `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_ diff --git a/docs/source/auto_examples/plot_fgw.ipynb b/docs/source/auto_examples/plot_fgw.ipynb new file mode 100644 index 0000000..1b150bd --- /dev/null +++ b/docs/source/auto_examples/plot_fgw.ipynb @@ -0,0 +1,162 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n# Plot Fused-gromov-Wasserstein\n\n\nThis example illustrates the computation of FGW for 1D measures[18].\n\n.. [18] Vayer Titouan, Chapel Laetitia, Flamary R{'e}mi, Tavenard Romain\n and Courty Nicolas\n \"Optimal Transport for structured data with application on graphs\"\n International Conference on Machine Learning (ICML). 2019.\n\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Author: Titouan Vayer <titouan.vayer@irisa.fr>\n#\n# License: MIT License\n\nimport matplotlib.pyplot as pl\nimport numpy as np\nimport ot\nfrom ot.gromov import gromov_wasserstein, fused_gromov_wasserstein" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Generate data\n---------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#%% parameters\n# We create two 1D random measures\nn = 20 # number of points in the first distribution\nn2 = 30 # number of points in the second distribution\nsig = 1 # std of first distribution\nsig2 = 0.1 # std of second distribution\n\nnp.random.seed(0)\n\nphi = np.arange(n)[:, None]\nxs = phi + sig * np.random.randn(n, 1)\nys = np.vstack((np.ones((n // 2, 1)), 0 * np.ones((n // 2, 1)))) + sig2 * np.random.randn(n, 1)\n\nphi2 = np.arange(n2)[:, None]\nxt = phi2 + sig * np.random.randn(n2, 1)\nyt = np.vstack((np.ones((n2 // 2, 1)), 0 * np.ones((n2 // 2, 1)))) + sig2 * np.random.randn(n2, 1)\nyt = yt[::-1, :]\n\np = ot.unif(n)\nq = ot.unif(n2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Plot data\n---------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#%% plot the distributions\n\npl.close(10)\npl.figure(10, (7, 7))\n\npl.subplot(2, 1, 1)\n\npl.scatter(ys, xs, c=phi, s=70)\npl.ylabel('Feature value a', fontsize=20)\npl.title('$\\mu=\\sum_i \\delta_{x_i,a_i}$', fontsize=25, usetex=True, y=1)\npl.xticks(())\npl.yticks(())\npl.subplot(2, 1, 2)\npl.scatter(yt, xt, c=phi2, s=70)\npl.xlabel('coordinates x/y', fontsize=25)\npl.ylabel('Feature value b', fontsize=20)\npl.title('$\\\\nu=\\sum_j \\delta_{y_j,b_j}$', fontsize=25, usetex=True, y=1)\npl.yticks(())\npl.tight_layout()\npl.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Create structure matrices and across-feature distance matrix\n---------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#%% Structure matrices and across-features distance matrix\nC1 = ot.dist(xs)\nC2 = ot.dist(xt)\nM = ot.dist(ys, yt)\nw1 = ot.unif(C1.shape[0])\nw2 = ot.unif(C2.shape[0])\nGot = ot.emd([], [], M)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Plot matrices\n---------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#%%\ncmap = 'Reds'\npl.close(10)\npl.figure(10, (5, 5))\nfs = 15\nl_x = [0, 5, 10, 15]\nl_y = [0, 5, 10, 15, 20, 25]\ngs = pl.GridSpec(5, 5)\n\nax1 = pl.subplot(gs[3:, :2])\n\npl.imshow(C1, cmap=cmap, interpolation='nearest')\npl.title(\"$C_1$\", fontsize=fs)\npl.xlabel(\"$k$\", fontsize=fs)\npl.ylabel(\"$i$\", fontsize=fs)\npl.xticks(l_x)\npl.yticks(l_x)\n\nax2 = pl.subplot(gs[:3, 2:])\n\npl.imshow(C2, cmap=cmap, interpolation='nearest')\npl.title(\"$C_2$\", fontsize=fs)\npl.ylabel(\"$l$\", fontsize=fs)\n#pl.ylabel(\"$l$\",fontsize=fs)\npl.xticks(())\npl.yticks(l_y)\nax2.set_aspect('auto')\n\nax3 = pl.subplot(gs[3:, 2:], sharex=ax2, sharey=ax1)\npl.imshow(M, cmap=cmap, interpolation='nearest')\npl.yticks(l_x)\npl.xticks(l_y)\npl.ylabel(\"$i$\", fontsize=fs)\npl.title(\"$M_{AB}$\", fontsize=fs)\npl.xlabel(\"$j$\", fontsize=fs)\npl.tight_layout()\nax3.set_aspect('auto')\npl.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Compute FGW/GW\n---------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#%% Computing FGW and GW\nalpha = 1e-3\n\not.tic()\nGwg, logw = fused_gromov_wasserstein(M, C1, C2, p, q, loss_fun='square_loss', alpha=alpha, verbose=True, log=True)\not.toc()\n\n#%reload_ext WGW\nGg, log = gromov_wasserstein(C1, C2, p, q, loss_fun='square_loss', verbose=True, log=True)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Visualize transport matrices\n---------\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#%% visu OT matrix\ncmap = 'Blues'\nfs = 15\npl.figure(2, (13, 5))\npl.clf()\npl.subplot(1, 3, 1)\npl.imshow(Got, cmap=cmap, interpolation='nearest')\n#pl.xlabel(\"$y$\",fontsize=fs)\npl.ylabel(\"$i$\", fontsize=fs)\npl.xticks(())\n\npl.title('Wasserstein ($M$ only)')\n\npl.subplot(1, 3, 2)\npl.imshow(Gg, cmap=cmap, interpolation='nearest')\npl.title('Gromov ($C_1,C_2$ only)')\npl.xticks(())\npl.subplot(1, 3, 3)\npl.imshow(Gwg, cmap=cmap, interpolation='nearest')\npl.title('FGW ($M+C_1,C_2$)')\n\npl.xlabel(\"$j$\", fontsize=fs)\npl.ylabel(\"$i$\", fontsize=fs)\n\npl.tight_layout()\npl.show()" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.6.8" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +}
\ No newline at end of file diff --git a/docs/source/auto_examples/plot_fgw.py b/docs/source/auto_examples/plot_fgw.py new file mode 100644 index 0000000..43efc94 --- /dev/null +++ b/docs/source/auto_examples/plot_fgw.py @@ -0,0 +1,173 @@ +# -*- coding: utf-8 -*- +""" +============================== +Plot Fused-gromov-Wasserstein +============================== + +This example illustrates the computation of FGW for 1D measures[18]. + +.. [18] Vayer Titouan, Chapel Laetitia, Flamary R{\'e}mi, Tavenard Romain + and Courty Nicolas + "Optimal Transport for structured data with application on graphs" + International Conference on Machine Learning (ICML). 2019. + +""" + +# Author: Titouan Vayer <titouan.vayer@irisa.fr> +# +# License: MIT License + +import matplotlib.pyplot as pl +import numpy as np +import ot +from ot.gromov import gromov_wasserstein, fused_gromov_wasserstein + +############################################################################## +# Generate data +# --------- + +#%% parameters +# We create two 1D random measures +n = 20 # number of points in the first distribution +n2 = 30 # number of points in the second distribution +sig = 1 # std of first distribution +sig2 = 0.1 # std of second distribution + +np.random.seed(0) + +phi = np.arange(n)[:, None] +xs = phi + sig * np.random.randn(n, 1) +ys = np.vstack((np.ones((n // 2, 1)), 0 * np.ones((n // 2, 1)))) + sig2 * np.random.randn(n, 1) + +phi2 = np.arange(n2)[:, None] +xt = phi2 + sig * np.random.randn(n2, 1) +yt = np.vstack((np.ones((n2 // 2, 1)), 0 * np.ones((n2 // 2, 1)))) + sig2 * np.random.randn(n2, 1) +yt = yt[::-1, :] + +p = ot.unif(n) +q = ot.unif(n2) + +############################################################################## +# Plot data +# --------- + +#%% plot the distributions + +pl.close(10) +pl.figure(10, (7, 7)) + +pl.subplot(2, 1, 1) + +pl.scatter(ys, xs, c=phi, s=70) +pl.ylabel('Feature value a', fontsize=20) +pl.title('$\mu=\sum_i \delta_{x_i,a_i}$', fontsize=25, usetex=True, y=1) +pl.xticks(()) +pl.yticks(()) +pl.subplot(2, 1, 2) +pl.scatter(yt, xt, c=phi2, s=70) +pl.xlabel('coordinates x/y', fontsize=25) +pl.ylabel('Feature value b', fontsize=20) +pl.title('$\\nu=\sum_j \delta_{y_j,b_j}$', fontsize=25, usetex=True, y=1) +pl.yticks(()) +pl.tight_layout() +pl.show() + +############################################################################## +# Create structure matrices and across-feature distance matrix +# --------- + +#%% Structure matrices and across-features distance matrix +C1 = ot.dist(xs) +C2 = ot.dist(xt) +M = ot.dist(ys, yt) +w1 = ot.unif(C1.shape[0]) +w2 = ot.unif(C2.shape[0]) +Got = ot.emd([], [], M) + +############################################################################## +# Plot matrices +# --------- + +#%% +cmap = 'Reds' +pl.close(10) +pl.figure(10, (5, 5)) +fs = 15 +l_x = [0, 5, 10, 15] +l_y = [0, 5, 10, 15, 20, 25] +gs = pl.GridSpec(5, 5) + +ax1 = pl.subplot(gs[3:, :2]) + +pl.imshow(C1, cmap=cmap, interpolation='nearest') +pl.title("$C_1$", fontsize=fs) +pl.xlabel("$k$", fontsize=fs) +pl.ylabel("$i$", fontsize=fs) +pl.xticks(l_x) +pl.yticks(l_x) + +ax2 = pl.subplot(gs[:3, 2:]) + +pl.imshow(C2, cmap=cmap, interpolation='nearest') +pl.title("$C_2$", fontsize=fs) +pl.ylabel("$l$", fontsize=fs) +#pl.ylabel("$l$",fontsize=fs) +pl.xticks(()) +pl.yticks(l_y) +ax2.set_aspect('auto') + +ax3 = pl.subplot(gs[3:, 2:], sharex=ax2, sharey=ax1) +pl.imshow(M, cmap=cmap, interpolation='nearest') +pl.yticks(l_x) +pl.xticks(l_y) +pl.ylabel("$i$", fontsize=fs) +pl.title("$M_{AB}$", fontsize=fs) +pl.xlabel("$j$", fontsize=fs) +pl.tight_layout() +ax3.set_aspect('auto') +pl.show() + +############################################################################## +# Compute FGW/GW +# --------- + +#%% Computing FGW and GW +alpha = 1e-3 + +ot.tic() +Gwg, logw = fused_gromov_wasserstein(M, C1, C2, p, q, loss_fun='square_loss', alpha=alpha, verbose=True, log=True) +ot.toc() + +#%reload_ext WGW +Gg, log = gromov_wasserstein(C1, C2, p, q, loss_fun='square_loss', verbose=True, log=True) + +############################################################################## +# Visualize transport matrices +# --------- + +#%% visu OT matrix +cmap = 'Blues' +fs = 15 +pl.figure(2, (13, 5)) +pl.clf() +pl.subplot(1, 3, 1) +pl.imshow(Got, cmap=cmap, interpolation='nearest') +#pl.xlabel("$y$",fontsize=fs) +pl.ylabel("$i$", fontsize=fs) +pl.xticks(()) + +pl.title('Wasserstein ($M$ only)') + +pl.subplot(1, 3, 2) +pl.imshow(Gg, cmap=cmap, interpolation='nearest') +pl.title('Gromov ($C_1,C_2$ only)') +pl.xticks(()) +pl.subplot(1, 3, 3) +pl.imshow(Gwg, cmap=cmap, interpolation='nearest') +pl.title('FGW ($M+C_1,C_2$)') + +pl.xlabel("$j$", fontsize=fs) +pl.ylabel("$i$", fontsize=fs) + +pl.tight_layout() +pl.show() diff --git a/docs/source/auto_examples/plot_fgw.rst b/docs/source/auto_examples/plot_fgw.rst new file mode 100644 index 0000000..aec725d --- /dev/null +++ b/docs/source/auto_examples/plot_fgw.rst @@ -0,0 +1,297 @@ + + +.. _sphx_glr_auto_examples_plot_fgw.py: + + +============================== +Plot Fused-gromov-Wasserstein +============================== + +This example illustrates the computation of FGW for 1D measures[18]. + +.. [18] Vayer Titouan, Chapel Laetitia, Flamary R{'e}mi, Tavenard Romain + and Courty Nicolas + "Optimal Transport for structured data with application on graphs" + International Conference on Machine Learning (ICML). 2019. + + + + +.. code-block:: python + + + # Author: Titouan Vayer <titouan.vayer@irisa.fr> + # + # License: MIT License + + import matplotlib.pyplot as pl + import numpy as np + import ot + from ot.gromov import gromov_wasserstein, fused_gromov_wasserstein + + + + + + + +Generate data +--------- + + + +.. code-block:: python + + + #%% parameters + # We create two 1D random measures + n = 20 # number of points in the first distribution + n2 = 30 # number of points in the second distribution + sig = 1 # std of first distribution + sig2 = 0.1 # std of second distribution + + np.random.seed(0) + + phi = np.arange(n)[:, None] + xs = phi + sig * np.random.randn(n, 1) + ys = np.vstack((np.ones((n // 2, 1)), 0 * np.ones((n // 2, 1)))) + sig2 * np.random.randn(n, 1) + + phi2 = np.arange(n2)[:, None] + xt = phi2 + sig * np.random.randn(n2, 1) + yt = np.vstack((np.ones((n2 // 2, 1)), 0 * np.ones((n2 // 2, 1)))) + sig2 * np.random.randn(n2, 1) + yt = yt[::-1, :] + + p = ot.unif(n) + q = ot.unif(n2) + + + + + + + +Plot data +--------- + + + +.. code-block:: python + + + #%% plot the distributions + + pl.close(10) + pl.figure(10, (7, 7)) + + pl.subplot(2, 1, 1) + + pl.scatter(ys, xs, c=phi, s=70) + pl.ylabel('Feature value a', fontsize=20) + pl.title('$\mu=\sum_i \delta_{x_i,a_i}$', fontsize=25, usetex=True, y=1) + pl.xticks(()) + pl.yticks(()) + pl.subplot(2, 1, 2) + pl.scatter(yt, xt, c=phi2, s=70) + pl.xlabel('coordinates x/y', fontsize=25) + pl.ylabel('Feature value b', fontsize=20) + pl.title('$\\nu=\sum_j \delta_{y_j,b_j}$', fontsize=25, usetex=True, y=1) + pl.yticks(()) + pl.tight_layout() + pl.show() + + + + +.. image:: /auto_examples/images/sphx_glr_plot_fgw_010.png + :align: center + + + + +Create structure matrices and across-feature distance matrix +--------- + + + +.. code-block:: python + + + #%% Structure matrices and across-features distance matrix + C1 = ot.dist(xs) + C2 = ot.dist(xt) + M = ot.dist(ys, yt) + w1 = ot.unif(C1.shape[0]) + w2 = ot.unif(C2.shape[0]) + Got = ot.emd([], [], M) + + + + + + + +Plot matrices +--------- + + + +.. code-block:: python + + + #%% + cmap = 'Reds' + pl.close(10) + pl.figure(10, (5, 5)) + fs = 15 + l_x = [0, 5, 10, 15] + l_y = [0, 5, 10, 15, 20, 25] + gs = pl.GridSpec(5, 5) + + ax1 = pl.subplot(gs[3:, :2]) + + pl.imshow(C1, cmap=cmap, interpolation='nearest') + pl.title("$C_1$", fontsize=fs) + pl.xlabel("$k$", fontsize=fs) + pl.ylabel("$i$", fontsize=fs) + pl.xticks(l_x) + pl.yticks(l_x) + + ax2 = pl.subplot(gs[:3, 2:]) + + pl.imshow(C2, cmap=cmap, interpolation='nearest') + pl.title("$C_2$", fontsize=fs) + pl.ylabel("$l$", fontsize=fs) + #pl.ylabel("$l$",fontsize=fs) + pl.xticks(()) + pl.yticks(l_y) + ax2.set_aspect('auto') + + ax3 = pl.subplot(gs[3:, 2:], sharex=ax2, sharey=ax1) + pl.imshow(M, cmap=cmap, interpolation='nearest') + pl.yticks(l_x) + pl.xticks(l_y) + pl.ylabel("$i$", fontsize=fs) + pl.title("$M_{AB}$", fontsize=fs) + pl.xlabel("$j$", fontsize=fs) + pl.tight_layout() + ax3.set_aspect('auto') + pl.show() + + + + +.. image:: /auto_examples/images/sphx_glr_plot_fgw_011.png + :align: center + + + + +Compute FGW/GW +--------- + + + +.. code-block:: python + + + #%% Computing FGW and GW + alpha = 1e-3 + + ot.tic() + Gwg, logw = fused_gromov_wasserstein(M, C1, C2, p, q, loss_fun='square_loss', alpha=alpha, verbose=True, log=True) + ot.toc() + + #%reload_ext WGW + Gg, log = gromov_wasserstein(C1, C2, p, q, loss_fun='square_loss', verbose=True, log=True) + + + + + +.. rst-class:: sphx-glr-script-out + + Out:: + + It. |Loss |Relative loss|Absolute loss + ------------------------------------------------ + 0|4.734462e+01|0.000000e+00|0.000000e+00 + 1|2.508258e+01|8.875498e-01|2.226204e+01 + 2|2.189329e+01|1.456747e-01|3.189297e+00 + 3|2.189329e+01|0.000000e+00|0.000000e+00 + Elapsed time : 0.0016989707946777344 s + It. |Loss |Relative loss|Absolute loss + ------------------------------------------------ + 0|4.683978e+04|0.000000e+00|0.000000e+00 + 1|3.860061e+04|2.134468e-01|8.239175e+03 + 2|2.182948e+04|7.682787e-01|1.677113e+04 + 3|2.182948e+04|0.000000e+00|0.000000e+00 + + +Visualize transport matrices +--------- + + + +.. code-block:: python + + + #%% visu OT matrix + cmap = 'Blues' + fs = 15 + pl.figure(2, (13, 5)) + pl.clf() + pl.subplot(1, 3, 1) + pl.imshow(Got, cmap=cmap, interpolation='nearest') + #pl.xlabel("$y$",fontsize=fs) + pl.ylabel("$i$", fontsize=fs) + pl.xticks(()) + + pl.title('Wasserstein ($M$ only)') + + pl.subplot(1, 3, 2) + pl.imshow(Gg, cmap=cmap, interpolation='nearest') + pl.title('Gromov ($C_1,C_2$ only)') + pl.xticks(()) + pl.subplot(1, 3, 3) + pl.imshow(Gwg, cmap=cmap, interpolation='nearest') + pl.title('FGW ($M+C_1,C_2$)') + + pl.xlabel("$j$", fontsize=fs) + pl.ylabel("$i$", fontsize=fs) + + pl.tight_layout() + pl.show() + + + +.. image:: /auto_examples/images/sphx_glr_plot_fgw_004.png + :align: center + + + + +**Total running time of the script:** ( 0 minutes 1.468 seconds) + + + +.. only :: html + + .. container:: sphx-glr-footer + + + .. container:: sphx-glr-download + + :download:`Download Python source code: plot_fgw.py <plot_fgw.py>` + + + + .. container:: sphx-glr-download + + :download:`Download Jupyter notebook: plot_fgw.ipynb <plot_fgw.ipynb>` + + +.. only:: html + + .. rst-class:: sphx-glr-signature + + `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_ diff --git a/docs/source/conf.py b/docs/source/conf.py index 433eca6..d29b829 100644 --- a/docs/source/conf.py +++ b/docs/source/conf.py @@ -15,7 +15,10 @@ import sys import os import re -import sphinx_gallery +try: + import sphinx_gallery +except ImportError: + print("warning sphinx-gallery not installed") # !!!! allow readthedoc compilation try: @@ -65,6 +68,8 @@ extensions = [ #'sphinx_gallery.gen_gallery', ] +napoleon_numpy_docstring = True + # Add any paths that contain templates here, relative to this directory. templates_path = ['_templates'] @@ -81,7 +86,7 @@ master_doc = 'index' # General information about the project. project = u'POT Python Optimal Transport' -copyright = u'2016-2018, Rémi Flamary, Nicolas Courty' +copyright = u'2016-2019, Rémi Flamary, Nicolas Courty' author = u'Rémi Flamary, Nicolas Courty' # The version info for the project you're documenting, acts as replacement for @@ -323,7 +328,10 @@ texinfo_documents = [ # Example configuration for intersphinx: refer to the Python standard library. -intersphinx_mapping = {'https://docs.python.org/': None} +intersphinx_mapping = {'python': ('https://docs.python.org/3', None), + 'numpy': ('http://docs.scipy.org/doc/numpy/', None), + 'scipy': ('http://docs.scipy.org/doc/scipy/reference/', None), + 'matplotlib': ('http://matplotlib.sourceforge.net/', None)} sphinx_gallery_conf = { 'examples_dirs': ['../../examples','../../examples/da'], diff --git a/docs/source/index.rst b/docs/source/index.rst index b8eabcb..9078d35 100644 --- a/docs/source/index.rst +++ b/docs/source/index.rst @@ -10,9 +10,10 @@ Contents -------- .. toctree:: - :maxdepth: 3 + :maxdepth: 2 self + quickstart all auto_examples/index diff --git a/docs/source/quickstart.rst b/docs/source/quickstart.rst new file mode 100644 index 0000000..1640d6a --- /dev/null +++ b/docs/source/quickstart.rst @@ -0,0 +1,877 @@ + +Quick start guide +================= + +In the following we provide some pointers about which functions and classes +to use for different problems related to optimal transport (OT) and machine +learning. We refer when we can to concrete examples in the documentation that +are also available as notebooks on the POT Github. + +This document is not a tutorial on numerical optimal transport. For this we strongly +recommend to read the very nice book [15]_ . + + +Optimal transport and Wasserstein distance +------------------------------------------ + +.. note:: + In POT, most functions that solve OT or regularized OT problems have two + versions that return the OT matrix or the value of the optimal solution. For + instance :any:`ot.emd` return the OT matrix and :any:`ot.emd2` return the + Wassertsein distance. This approach has been implemented in practice for all + solvers that return an OT matrix (even Gromov-Wasserstsein) + +Solving optimal transport +^^^^^^^^^^^^^^^^^^^^^^^^^ + +The optimal transport problem between discrete distributions is often expressed +as + +.. math:: + \gamma^* = arg\min_\gamma \quad \sum_{i,j}\gamma_{i,j}M_{i,j} + + s.t. \gamma 1 = a; \gamma^T 1= b; \gamma\geq 0 + +where : + +- :math:`M\in\mathbb{R}_+^{m\times n}` is the metric cost matrix defining the cost to move mass from bin :math:`a_i` to bin :math:`b_j`. +- :math:`a` and :math:`b` are histograms on the simplex (positive, sum to 1) that represent the +weights of each samples in the source an target distributions. + +Solving the linear program above can be done using the function :any:`ot.emd` +that will return the optimal transport matrix :math:`\gamma^*`: + +.. code:: python + + # a,b are 1D histograms (sum to 1 and positive) + # M is the ground cost matrix + T=ot.emd(a,b,M) # exact linear program + +The method implemented for solving the OT problem is the network simplex, it is +implemented in C from [1]_. It has a complexity of :math:`O(n^3)` but the +solver is quite efficient and uses sparsity of the solution. + +.. hint:: + Examples of use for :any:`ot.emd` are available in : + + - :any:`auto_examples/plot_OT_2D_samples` + - :any:`auto_examples/plot_OT_1D` + - :any:`auto_examples/plot_OT_L1_vs_L2` + + +Computing Wasserstein distance +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The value of the OT solution is often more of interest than the OT matrix : + +.. math:: + OT(a,b)=\min_\gamma \quad \sum_{i,j}\gamma_{i,j}M_{i,j} + + s.t. \gamma 1 = a; \gamma^T 1= b; \gamma\geq 0 + + +It can computed from an already estimated OT matrix with +:code:`np.sum(T*M)` or directly with the function :any:`ot.emd2`. + +.. code:: python + + # a,b are 1D histograms (sum to 1 and positive) + # M is the ground cost matrix + W=ot.emd2(a,b,M) # Wasserstein distance / EMD value + +Note that the well known `Wasserstein distance +<https://en.wikipedia.org/wiki/Wasserstein_metric>`_ between distributions a and +b is defined as + + + .. math:: + + W_p(a,b)=(\min_\gamma \sum_{i,j}\gamma_{i,j}\|x_i-y_j\|_p)^\frac{1}{p} + + s.t. \gamma 1 = a; \gamma^T 1= b; \gamma\geq 0 + +This means that if you want to compute the :math:`W_2` you need to compute the +square root of :any:`ot.emd2` when providing +:code:`M=ot.dist(xs,xt)` that use the squared euclidean distance by default. Computing +the :math:`W_1` wasserstein distance can be done directly with :any:`ot.emd2` +when providing :code:`M=ot.dist(xs,xt, metric='euclidean')` to use the euclidean +distance. + + +.. hint:: + An example of use for :any:`ot.emd2` is available in : + + - :any:`auto_examples/plot_compute_emd` + + +Special cases +^^^^^^^^^^^^^ + +Note that the OT problem and the corresponding Wasserstein distance can in some +special cases be computed very efficiently. + +For instance when the samples are in 1D, then the OT problem can be solved in +:math:`O(n\log(n))` by using a simple sorting. In this case we provide the +function :any:`ot.emd_1d` and :any:`ot.emd2_1d` to return respectively the OT +matrix and value. Note that since the solution is very sparse the :code:`sparse` +parameter of :any:`ot.emd_1d` allows for solving and returning the solution for +very large problems. Note that in order to compute directly the :math:`W_p` +Wasserstein distance in 1D we provide the function :any:`ot.wasserstein_1d` that +takes :code:`p` as a parameter. + +Another special case for estimating OT and Monge mapping is between Gaussian +distributions. In this case there exists a close form solution given in Remark +2.29 in [15]_ and the Monge mapping is an affine function and can be +also computed from the covariances and means of the source and target +distributions. In the case when the finite sample dataset is supposed gaussian, we provide +:any:`ot.da.OT_mapping_linear` that returns the parameters for the Monge +mapping. + + +Regularized Optimal Transport +----------------------------- + +Recent developments have shown the interest of regularized OT both in terms of +computational and statistical properties. +We address in this section the regularized OT problems that can be expressed as + +.. math:: + \gamma^* = arg\min_\gamma \quad \sum_{i,j}\gamma_{i,j}M_{i,j} + \lambda\Omega(\gamma) + + s.t. \gamma 1 = a; \gamma^T 1= b; \gamma\geq 0 + + +where : + +- :math:`M\in\mathbb{R}_+^{m\times n}` is the metric cost matrix defining the cost to move mass from bin :math:`a_i` to bin :math:`b_j`. +- :math:`a` and :math:`b` are histograms (positive, sum to 1) that represent the weights of each samples in the source an target distributions. +- :math:`\Omega` is the regularization term. + +We discuss in the following specific algorithms that can be used depending on +the regularization term. + + +Entropic regularized OT +^^^^^^^^^^^^^^^^^^^^^^^ + +This is the most common regularization used for optimal transport. It has been +proposed in the ML community by Marco Cuturi in his seminal paper [2]_. This +regularization has the following expression + +.. math:: + \Omega(\gamma)=\sum_{i,j}\gamma_{i,j}\log(\gamma_{i,j}) + + +The use of the regularization term above in the optimization problem has a very +strong impact. First it makes the problem smooth which leads to new optimization +procedures such as the well known Sinkhorn algorithm [2]_ or L-BFGS (see +:any:`ot.smooth` ). Next it makes the problem +strictly convex meaning that there will be a unique solution. Finally the +solution of the resulting optimization problem can be expressed as: + +.. math:: + + \gamma_\lambda^*=\text{diag}(u)K\text{diag}(v) + +where :math:`u` and :math:`v` are vectors and :math:`K=\exp(-M/\lambda)` where +the :math:`\exp` is taken component-wise. In order to solve the optimization +problem, on can use an alternative projection algorithm called Sinkhorn-Knopp that can be very +efficient for large values if regularization. + +The Sinkhorn-Knopp algorithm is implemented in :any:`ot.sinkhorn` and +:any:`ot.sinkhorn2` that return respectively the OT matrix and the value of the +linear term. Note that the regularization parameter :math:`\lambda` in the +equation above is given to those functions with the parameter :code:`reg`. + + >>> import ot + >>> a=[.5,.5] + >>> b=[.5,.5] + >>> M=[[0.,1.],[1.,0.]] + >>> ot.sinkhorn(a,b,M,1) + array([[ 0.36552929, 0.13447071], + [ 0.13447071, 0.36552929]]) + +More details about the algorithms used are given in the following note. + +.. note:: + The main function to solve entropic regularized OT is :any:`ot.sinkhorn`. + This function is a wrapper and the parameter :code:`method` help you select + the actual algorithm used to solve the problem: + + + :code:`method='sinkhorn'` calls :any:`ot.bregman.sinkhorn_knopp` the + classic algorithm [2]_. + + :code:`method='sinkhorn_stabilized'` calls :any:`ot.bregman.sinkhorn_stabilized` the + log stabilized version of the algorithm [9]_. + + :code:`method='sinkhorn_epsilon_scaling'` calls + :any:`ot.bregman.sinkhorn_epsilon_scaling` the epsilon scaling version + of the algorithm [9]_. + + :code:`method='greenkhorn'` calls :any:`ot.bregman.greenkhorn` the + greedy sinkhorn verison of the algorithm [22]_. + + In addition to all those variants of sinkhorn, we have another + implementation solving the problem in the smooth dual or semi-dual in + :any:`ot.smooth`. This solver uses the :any:`scipy.optimize.minimize` + function to solve the smooth problem with :code:`L-BFGS-B` algorithm. Tu use + this solver, use functions :any:`ot.smooth.smooth_ot_dual` or + :any:`ot.smooth.smooth_ot_semi_dual` with parameter :code:`reg_type='kl'` to + choose entropic/Kullbach Leibler regularization. + + +Recently [23]_ introduced the sinkhorn divergence that build from entropic +regularization to compute fast and differentiable geometric divergence between +empirical distributions. Note that we provide a function that compute directly +(with no need to pre compute the :code:`M` matrix) +the sinkhorn divergence for empirical distributions in +:any:`ot.bregman.empirical_sinkhorn_divergence`. Similarly one can compute the +OT matrix and loss for empirical distributions with respectively +:any:`ot.bregman.empirical_sinkhorn` and :any:`ot.bregman.empirical_sinkhorn2`. + + +Finally note that we also provide in :any:`ot.stochastic` several implementation +of stochastic solvers for entropic regularized OT [18]_ [19]_. Those pure Python +implementations are not optimized for speed but provide a roust implementation +of algorithms in [18]_ [19]_. + +.. hint:: + Examples of use for :any:`ot.sinkhorn` are available in : + + - :any:`auto_examples/plot_OT_2D_samples` + - :any:`auto_examples/plot_OT_1D` + - :any:`auto_examples/plot_OT_1D_smooth` + - :any:`auto_examples/plot_stochastic` + + +Other regularization +^^^^^^^^^^^^^^^^^^^^ + +While entropic OT is the most common and favored in practice, there exist other +kind of regularization. We provide in POT two specific solvers for other +regularization terms, namely quadratic regularization and group lasso +regularization. But we also provide in :any:`ot.optim` two generic solvers that allows solving any +smooth regularization in practice. + +Quadratic regularization +"""""""""""""""""""""""" + +The first general regularization term we can solve is the quadratic +regularization of the form + +.. math:: + \Omega(\gamma)=\sum_{i,j} \gamma_{i,j}^2 + +this regularization term has a similar effect to entropic regularization in +densifying the OT matrix but it keeps some sort of sparsity that is lost with +entropic regularization as soon as :math:`\lambda>0` [17]_. This problem can be +solved with POT using solvers from :any:`ot.smooth`, more specifically +functions :any:`ot.smooth.smooth_ot_dual` or +:any:`ot.smooth.smooth_ot_semi_dual` with parameter :code:`reg_type='l2'` to +choose the quadratic regularization. + +.. hint:: + Examples of quadratic regularization are available in : + + - :any:`auto_examples/plot_OT_1D_smooth` + - :any:`auto_examples/plot_optim_OTreg` + + + +Group Lasso regularization +"""""""""""""""""""""""""" + +Another regularization that has been used in recent years [5]_ is the group lasso +regularization + +.. math:: + \Omega(\gamma)=\sum_{j,G\in\mathcal{G}} \|\gamma_{G,j}\|_q^p + +where :math:`\mathcal{G}` contains non overlapping groups of lines in the OT +matrix. This regularization proposed in [5]_ will promote sparsity at the group level and for +instance will force target samples to get mass from a small number of groups. +Note that the exact OT solution is already sparse so this regularization does +not make sens if it is not combined with entropic regularization. Depending on +the choice of :code:`p` and :code:`q`, the problem can be solved with different +approaches. When :code:`q=1` and :code:`p<1` the problem is non convex but can +be solved using an efficient majoration minimization approach with +:any:`ot.sinkhorn_lpl1_mm`. When :code:`q=2` and :code:`p=1` we recover the +convex group lasso and we provide a solver using generalized conditional +gradient algorithm [7]_ in function +:any:`ot.da.sinkhorn_l1l2_gl`. + +.. hint:: + Examples of group Lasso regularization are available in : + + - :any:`auto_examples/plot_otda_classes` + - :any:`auto_examples/plot_otda_d2` + + +Generic solvers +""""""""""""""" + +Finally we propose in POT generic solvers that can be used to solve any +regularization as long as you can provide a function computing the +regularization and a function computing its gradient (or sub-gradient). + +In order to solve + +.. math:: + \gamma^* = arg\min_\gamma \quad \sum_{i,j}\gamma_{i,j}M_{i,j} + \lambda\Omega(\gamma) + + s.t. \gamma 1 = a; \gamma^T 1= b; \gamma\geq 0 + +you can use function :any:`ot.optim.cg` that will use a conditional gradient as +proposed in [6]_ . You need to provide the regularization function as parameter +``f`` and its gradient as parameter ``df``. Note that the conditional gradient relies on +iterative solving of a linearization of the problem using the exact +:any:`ot.emd` so it can be slow in practice. But, being an interior point +algorithm, it always returns a +transport matrix that does not violates the marginals. + +Another generic solver is proposed to solve the problem + +.. math:: + \gamma^* = arg\min_\gamma \quad \sum_{i,j}\gamma_{i,j}M_{i,j}+ \lambda_e\Omega_e(\gamma) + \lambda\Omega(\gamma) + + s.t. \gamma 1 = a; \gamma^T 1= b; \gamma\geq 0 + +where :math:`\Omega_e` is the entropic regularization. In this case we use a +generalized conditional gradient [7]_ implemented in :any:`ot.optim.gcg` that +does not linearize the entropic term but +relies on :any:`ot.sinkhorn` for its iterations. + +.. hint:: + An example of generic solvers are available in : + + - :any:`auto_examples/plot_optim_OTreg` + + +Wasserstein Barycenters +----------------------- + +A Wasserstein barycenter is a distribution that minimize its Wasserstein +distance with respect to other distributions [16]_. It corresponds to minimizing the +following problem by searching a distribution :math:`\mu` such that + +.. math:: + \min_\mu \quad \sum_{k} w_kW(\mu,\mu_k) + + +In practice we model a distribution with a finite number of support position: + +.. math:: + \mu=\sum_{i=1}^n a_i\delta_{x_i} + +where :math:`a` is an histogram on the simplex and the :math:`\{x_i\}` are the +position of the support. We can clearly see here that optimizing :math:`\mu` can +be done by searching for optimal weights :math:`a` or optimal support +:math:`\{x_i\}` (optimizing both is also an option). +We provide in POT solvers to estimate a discrete +Wasserstein barycenter in both cases. + +Barycenters with fixed support +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +When optimizing a barycenter with a fixed support, the optimization problem can +be expressed as + +.. math:: + \min_a \quad \sum_{k} w_k W(a,b_k) + +where :math:`b_k` are also weights in the simplex. In the non-regularized case, +the problem above is a classical linear program. In this case we propose a +solver :any:`ot.lp.barycenter` that rely on generic LP solvers. By default the +function uses :any:`scipy.optimize.linprog`, but more efficient LP solvers from +cvxopt can be also used by changing parameter :code:`solver`. Note that this problem +requires to solve a very large linear program and can be very slow in +practice. + +Similarly to the OT problem, OT barycenters can be computed in the regularized +case. When using entropic regularization is used, the problem can be solved with a +generalization of the sinkhorn algorithm based on bregman projections [3]_. This +algorithm is provided in function :any:`ot.bregman.barycenter` also available as +:any:`ot.barycenter`. In this case, the algorithm scales better to large +distributions and rely only on matrix multiplications that can be performed in +parallel. + +In addition to the speedup brought by regularization, one can also greatly +accelerate the estimation of Wasserstein barycenter when the support has a +separable structure [21]_. In the case of 2D images for instance one can replace +the matrix vector production in the Bregman projections by convolution +operators. We provide an implementation of this algorithm in function +:any:`ot.bregman.convolutional_barycenter2d`. + +.. hint:: + Examples of Wasserstein (:any:`ot.lp.barycenter`) and regularized Wasserstein + barycenter (:any:`ot.bregman.barycenter`) computation are available in : + + - :any:`auto_examples/plot_barycenter_1D` + - :any:`auto_examples/plot_barycenter_lp_vs_entropic` + + An example of convolutional barycenter + (:any:`ot.bregman.convolutional_barycenter2d`) computation + for 2D images is available + in : + + - :any:`auto_examples/plot_convolutional_barycenter` + + + +Barycenters with free support +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Estimating the Wasserstein barycenter with free support but fixed weights +corresponds to solving the following optimization problem: + +.. math:: + \min_{\{x_i\}} \quad \sum_{k} w_kW(\mu,\mu_k) + + s.t. \quad \mu=\sum_{i=1}^n a_i\delta_{x_i} + +We provide a solver based on [20]_ in +:any:`ot.lp.free_support_barycenter`. This function minimize the problem and +return a locally optimal support :math:`\{x_i\}` for uniform or given weights +:math:`a`. + + .. hint:: + + An example of the free support barycenter estimation is available + in : + + - :any:`auto_examples/plot_free_support_barycenter` + + + + +Monge mapping and Domain adaptation +----------------------------------- + +The original transport problem investigated by Gaspard Monge was seeking for a +mapping function that maps (or transports) between a source and target +distribution but that minimizes the transport loss. The existence and uniqueness of this +optimal mapping is still an open problem in the general case but has been proven +for smooth distributions by Brenier in his eponym `theorem +<https://who.rocq.inria.fr/Jean-David.Benamou/demiheure.pdf>`__. We provide in +:any:`ot.da` several solvers for smooth Monge mapping estimation and domain +adaptation from discrete distributions. + +Monge Mapping estimation +^^^^^^^^^^^^^^^^^^^^^^^^ + +We now discuss several approaches that are implemented in POT to estimate or +approximate a Monge mapping from finite distributions. + +First note that when the source and target distributions are supposed to be Gaussian +distributions, there exists a close form solution for the mapping and its an +affine function [14]_ of the form :math:`T(x)=Ax+b` . In this case we provide the function +:any:`ot.da.OT_mapping_linear` that return the operator :math:`A` and vector +:math:`b`. Note that if the number of samples is too small there is a parameter +:code:`reg` that provide a regularization for the covariance matrix estimation. + +For a more general mapping estimation we also provide the barycentric mapping +proposed in [6]_ . It is implemented in the class :any:`ot.da.EMDTransport` and +other transport based classes in :any:`ot.da` . Those classes are discussed more +in the following but follow an interface similar to sklearn classes. Finally a +method proposed in [8]_ that estimates a continuous mapping approximating the +barycentric mapping is provided in :any:`ot.da.joint_OT_mapping_linear` for +linear mapping and :any:`ot.da.joint_OT_mapping_kernel` for non linear mapping. + + .. hint:: + + An example of the linear Monge mapping estimation is available + in : + + - :any:`auto_examples/plot_otda_linear_mapping` + +Domain adaptation classes +^^^^^^^^^^^^^^^^^^^^^^^^^ + +The use of OT for domain adaptation (OTDA) has been first proposed in [5]_ that also +introduced the group Lasso regularization. The main idea of OTDA is to estimate +a mapping of the samples between source and target distributions which allows to +transport labeled source samples onto the target distribution with no labels. + +We provide several classes based on :any:`ot.da.BaseTransport` that provide +several OT and mapping estimations. The interface of those classes is similar to +classifiers in sklearn toolbox. At initialization, several parameters such as + regularization parameter value can be set. Then one needs to estimate the +mapping with function :any:`ot.da.BaseTransport.fit`. Finally one can map the +samples from source to target with :any:`ot.da.BaseTransport.transform` and +from target to source with :any:`ot.da.BaseTransport.inverse_transform`. + +Here is +an example for class :any:`ot.da.EMDTransport` : + +.. code:: + + ot_emd = ot.da.EMDTransport() + ot_emd.fit(Xs=Xs, Xt=Xt) + + Mapped_Xs= ot_emd.transform(Xs=Xs) + +A list of the provided implementation is given in the following note. + +.. note:: + + Here is a list of the OT mapping classes inheriting from + :any:`ot.da.BaseTransport` + + * :any:`ot.da.EMDTransport` : Barycentric mapping with EMD transport + * :any:`ot.da.SinkhornTransport` : Barycentric mapping with Sinkhorn transport + * :any:`ot.da.SinkhornL1l2Transport` : Barycentric mapping with Sinkhorn + + group Lasso regularization [5]_ + * :any:`ot.da.SinkhornLpl1Transport` : Barycentric mapping with Sinkhorn + + non convex group Lasso regularization [5]_ + * :any:`ot.da.LinearTransport` : Linear mapping estimation between Gaussians + [14]_ + * :any:`ot.da.MappingTransport` : Nonlinear mapping estimation [8]_ + +.. hint:: + + Example of the use of OTDA classes are available in : + + - :any:`auto_examples/plot_otda_color_images` + - :any:`auto_examples/plot_otda_mapping` + - :any:`auto_examples/plot_otda_mapping_colors_images` + - :any:`auto_examples/plot_otda_semi_supervised` + +Other applications +------------------ + +We discuss in the following several OT related problems and tools that has been +proposed in the OT and machine learning community. + +Wasserstein Discriminant Analysis +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Wasserstein Discriminant Analysis [11]_ is a generalization of `Fisher Linear Discriminant +Analysis <https://en.wikipedia.org/wiki/Linear_discriminant_analysis>`__ that +allows discrimination between classes that are not linearly separable. It +consist in finding a linear projector optimizing the following criterion + +.. math:: + P = \text{arg}\min_P \frac{\sum_i OT_e(\mu_i\#P,\mu_i\#P)}{\sum_{i,j\neq i} + OT_e(\mu_i\#P,\mu_j\#P)} + +where :math:`\#` is the push-forward operator, :math:`OT_e` is the entropic OT +loss and :math:`\mu_i` is the +distribution of samples from class :math:`i`. :math:`P` is also constrained to +be in the Stiefel manifold. WDA can be solved in POT using function +:any:`ot.dr.wda`. It requires to have installed :code:`pymanopt` and +:code:`autograd` for manifold optimization and automatic differentiation +respectively. Note that we also provide the Fisher discriminant estimator in +:any:`ot.dr.fda` for easy comparison. + +.. warning:: + Note that due to the hard dependency on :code:`pymanopt` and + :code:`autograd`, :any:`ot.dr` is not imported by default. If you want to + use it you have to specifically import it with :code:`import ot.dr` . + +.. hint:: + + An example of the use of WDA is available in : + + - :any:`auto_examples/plot_WDA` + + +Unbalanced optimal transport +^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Unbalanced OT is a relaxation of the original OT problem where the violation of +the constraint on the marginals is added to the objective of the optimization +problem: + +.. math:: + \min_\gamma \quad \sum_{i,j}\gamma_{i,j}M_{i,j} + reg\cdot\Omega(\gamma) + \alpha KL(\gamma 1, a) + \alpha KL(\gamma^T 1, b) + + s.t. \quad \gamma\geq 0 + + +where KL is the Kullback-Leibler divergence. This formulation allows for +computing approximate mapping between distributions that do not have the same +amount of mass. Interestingly the problem can be solved with a generalization of +the Bregman projections algorithm [10]_. We provide a solver for unbalanced OT +in :any:`ot.unbalanced` and more specifically +in function :any:`ot.sinkhorn_unbalanced`. A solver for unbalanced OT barycenter +is available in :any:`ot.barycenter_unbalanced`. + + +.. hint:: + + Examples of the use of :any:`ot.sinkhorn_unbalanced` and + :any:`ot.barycenter_unbalanced` are available in : + + - :any:`auto_examples/plot_UOT_1D` + - :any:`auto_examples/plot_UOT_barycenter_1D` + + +Gromov-Wasserstein +^^^^^^^^^^^^^^^^^^ + +Gromov Wasserstein (GW) is a generalization of OT to distributions that do not lie in +the same space [13]_. In this case one cannot compute distance between samples +from the two distributions. [13]_ proposed instead to realign the metric spaces +by computing a transport between distance matrices. The Gromow Wasserstein +alignement between two distributions can be expressed as the one minimizing: + +.. math:: + GW = \min_\gamma \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*\gamma_{i,j}*\gamma_{k,l} + + s.t. \gamma 1 = a; \gamma^T 1= b; \gamma\geq 0 + +where ::math:`C1` is the distance matrix between samples in the source +distribution and :math:`C2` the one between samples in the target, +:math:`L(C1_{i,k},C2_{j,l})` is a measure of similarity between +:math:`C1_{i,k}` and :math:`C2_{j,l}` often chosen as +:math:`L(C1_{i,k},C2_{j,l})=\|C1_{i,k}-C2_{j,l}\|^2`. The optimization problem +above is a non-convex quadratic program but we provide a solver that finds a +local minimum using conditional gradient in :any:`ot.gromov.gromov_wasserstein`. +There also exists an entropic regularized variant of GW that has been proposed in +[12]_ and we provide an implementation of their algorithm in +:any:`ot.gromov.entropic_gromov_wasserstein`. + +Note that similarly to Wasserstein distance GW allows for the definition of GW +barycenters that can be expressed as + +.. math:: + \min_{C\geq 0} \quad \sum_{k} w_k GW(C,Ck) + +where :math:`Ck` is the distance matrix between samples in distribution +:math:`k`. Note that interestingly the barycenter is defined as a symmetric +positive matrix. We provide a block coordinate optimization procedure in +:any:`ot.gromov.gromov_barycenters` and +:any:`ot.gromov.entropic_gromov_barycenters` for non-regularized and regularized +barycenters respectively. + +Finally note that recently a fusion between Wasserstein and GW, coined Fused +Gromov-Wasserstein (FGW) has been proposed +in [24]_. It allows to compute a similarity between objects that are only partly in +the same space. As such it can be used to measure similarity between labeled +graphs for instance and also provide computable barycenters. +The implementations of FGW and FGW barycenter is provided in functions +:any:`ot.gromov.fused_gromov_wasserstein` and :any:`ot.gromov.fgw_barycenters`. + +.. hint:: + + Examples of computation of GW, regularized G and FGW are available in : + + - :any:`auto_examples/plot_gromov` + - :any:`auto_examples/plot_fgw` + + Examples of GW, regularized GW and FGW barycenters are available in : + + - :any:`auto_examples/plot_gromov_barycenter` + - :any:`auto_examples/plot_barycenter_fgw` + + +GPU acceleration +^^^^^^^^^^^^^^^^ + +We provide several implementation of our OT solvers in :any:`ot.gpu`. Those +implementations use the :code:`cupy` toolbox that obviously need to be installed. + + +.. note:: + + Several implementations of POT functions (mainly those relying on linear + algebra) have been implemented in :any:`ot.gpu`. Here is a short list on the + main entries: + + - :any:`ot.gpu.dist` : computation of distance matrix + - :any:`ot.gpu.sinkhorn` : computation of sinkhorn + - :any:`ot.gpu.sinkhorn_lpl1_mm` : computation of sinkhorn + group lasso + +Note that while the :any:`ot.gpu` module has been designed to be compatible with +POT, calling its function with :any:`numpy` arrays will incur a large overhead due to +the memory copy of the array on GPU prior to computation and conversion of the +array after computation. To avoid this overhead, we provide functions +:any:`ot.gpu.to_gpu` and :any:`ot.gpu.to_np` that perform the conversion +explicitly. + + +.. warning:: + Note that due to the hard dependency on :code:`cupy`, :any:`ot.gpu` is not + imported by default. If you want to + use it you have to specifically import it with :code:`import ot.gpu` . + + +FAQ +--- + + + +1. **How to solve a discrete optimal transport problem ?** + + The solver for discrete OT is the function :py:mod:`ot.emd` that returns + the OT transport matrix. If you want to solve a regularized OT you can + use :py:mod:`ot.sinkhorn`. + + + Here is a simple use case: + + .. code:: python + + # a,b are 1D histograms (sum to 1 and positive) + # M is the ground cost matrix + T=ot.emd(a,b,M) # exact linear program + T_reg=ot.sinkhorn(a,b,M,reg) # entropic regularized OT + + More detailed examples can be seen on this example: + :doc:`auto_examples/plot_OT_2D_samples` + + +2. **pip install POT fails with error : ImportError: No module named Cython.Build** + + As discussed shortly in the README file. POT requires to have :code:`numpy` + and :code:`cython` installed to build. This corner case is not yet handled + by :code:`pip` and for now you need to install both library prior to + installing POT. + + Note that this problem do not occur when using conda-forge since the packages + there are pre-compiled. + + See `Issue #59 <https://github.com/rflamary/POT/issues/59>`__ for more + details. + +3. **Why is Sinkhorn slower than EMD ?** + + This might come from the choice of the regularization term. The speed of + convergence of sinkhorn depends directly on this term [22]_ and when the + regularization gets very small the problem try and approximate the exact OT + which leads to slow convergence in addition to numerical problems. In other + words, for large regularization sinkhorn will be very fast to converge, for + small regularization (when you need an OT matrix close to the true OT), it + might be quicker to use the EMD solver. + + Also note that the numpy implementation of the sinkhorn can use parallel + computation depending on the configuration of your system but very important + speedup can be obtained by using a GPU implementation since all operations + are matrix/vector products. + +4. **Using GPU fails with error: module 'ot' has no attribute 'gpu'** + + In order to limit import time and hard dependencies in POT. we do not import + some sub-modules automatically with :code:`import ot`. In order to use the + acceleration in :any:`ot.gpu` you need first to import is with + :code:`import ot.gpu`. + + See `Issue #85 <https://github.com/rflamary/POT/issues/85>`__ and :any:`ot.gpu` + for more details. + + +References +---------- + +.. [1] Bonneel, N., Van De Panne, M., Paris, S., & Heidrich, W. (2011, + December). `Displacement nterpolation using Lagrangian mass transport + <https://people.csail.mit.edu/sparis/publi/2011/sigasia/Bonneel_11_Displacement_Interpolation.pdf>`__. + In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p. 158). ACM. + +.. [2] Cuturi, M. (2013). `Sinkhorn distances: Lightspeed computation of + optimal transport <https://arxiv.org/pdf/1306.0895.pdf>`__. In Advances + in Neural Information Processing Systems (pp. 2292-2300). + +.. [3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. + (2015). `Iterative Bregman projections for regularized transportation + problems <https://arxiv.org/pdf/1412.5154.pdf>`__. SIAM Journal on + Scientific Computing, 37(2), A1111-A1138. + +.. [4] S. Nakhostin, N. Courty, R. Flamary, D. Tuia, T. Corpetti, + `Supervised planetary unmixing with optimal + transport <https://hal.archives-ouvertes.fr/hal-01377236/document>`__, + Whorkshop on Hyperspectral Image and Signal Processing : Evolution in + Remote Sensing (WHISPERS), 2016. + +.. [5] N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, `Optimal Transport + for Domain Adaptation <https://arxiv.org/pdf/1507.00504.pdf>`__, in IEEE + Transactions on Pattern Analysis and Machine Intelligence , vol.PP, + no.99, pp.1-1 + +.. [6] Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014). + `Regularized discrete optimal + transport <https://arxiv.org/pdf/1307.5551.pdf>`__. SIAM Journal on + Imaging Sciences, 7(3), 1853-1882. + +.. [7] Rakotomamonjy, A., Flamary, R., & Courty, N. (2015). `Generalized + conditional gradient: analysis of convergence and + applications <https://arxiv.org/pdf/1510.06567.pdf>`__. arXiv preprint + arXiv:1510.06567. + +.. [8] M. Perrot, N. Courty, R. Flamary, A. Habrard (2016), `Mapping + estimation for discrete optimal + transport <http://remi.flamary.com/biblio/perrot2016mapping.pdf>`__, + Neural Information Processing Systems (NIPS). + +.. [9] Schmitzer, B. (2016). `Stabilized Sparse Scaling Algorithms for + Entropy Regularized Transport + Problems <https://arxiv.org/pdf/1610.06519.pdf>`__. arXiv preprint + arXiv:1610.06519. + +.. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). + `Scaling algorithms for unbalanced transport + problems <https://arxiv.org/pdf/1607.05816.pdf>`__. arXiv preprint + arXiv:1607.05816. + +.. [11] Flamary, R., Cuturi, M., Courty, N., & Rakotomamonjy, A. (2016). + `Wasserstein Discriminant + Analysis <https://arxiv.org/pdf/1608.08063.pdf>`__. arXiv preprint + arXiv:1608.08063. + +.. [12] Gabriel Peyré, Marco Cuturi, and Justin Solomon (2016), + `Gromov-Wasserstein averaging of kernel and distance + matrices <http://proceedings.mlr.press/v48/peyre16.html>`__ + International Conference on Machine Learning (ICML). + +.. [13] Mémoli, Facundo (2011). `Gromov–Wasserstein distances and the + metric approach to object + matching <https://media.adelaide.edu.au/acvt/Publications/2011/2011-Gromov%E2%80%93Wasserstein%20Distances%20and%20the%20Metric%20Approach%20to%20Object%20Matching.pdf>`__. + Foundations of computational mathematics 11.4 : 417-487. + +.. [14] Knott, M. and Smith, C. S. (1984). `On the optimal mapping of + distributions <https://link.springer.com/article/10.1007/BF00934745>`__, + Journal of Optimization Theory and Applications Vol 43. + +.. [15] Peyré, G., & Cuturi, M. (2018). `Computational Optimal + Transport <https://arxiv.org/pdf/1803.00567.pdf>`__ . + +.. [16] Agueh, M., & Carlier, G. (2011). `Barycenters in the Wasserstein + space <https://hal.archives-ouvertes.fr/hal-00637399/document>`__. SIAM + Journal on Mathematical Analysis, 43(2), 904-924. + +.. [17] Blondel, M., Seguy, V., & Rolet, A. (2018). `Smooth and Sparse + Optimal Transport <https://arxiv.org/abs/1710.06276>`__. Proceedings of + the Twenty-First International Conference on Artificial Intelligence and + Statistics (AISTATS). + +.. [18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016) `Stochastic + Optimization for Large-scale Optimal + Transport <https://arxiv.org/abs/1605.08527>`__. 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 <https://arxiv.org/pdf/1711.02283.pdf>`__. International + Conference on Learning Representation (2018) + +.. [20] Cuturi, M. and Doucet, A. (2014) `Fast Computation of Wasserstein + Barycenters <http://proceedings.mlr.press/v32/cuturi14.html>`__. + International Conference in Machine Learning + +.. [21] Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher, A., + Nguyen, A. & Guibas, L. (2015). `Convolutional wasserstein distances: + Efficient optimal transportation on geometric + domains <https://dl.acm.org/citation.cfm?id=2766963>`__. ACM + Transactions on Graphics (TOG), 34(4), 66. + +.. [22] J. Altschuler, J.Weed, P. Rigollet, (2017) `Near-linear time + approximation algorithms for optimal transport via Sinkhorn + iteration <https://papers.nips.cc/paper/6792-near-linear-time-approximation-algorithms-for-optimal-transport-via-sinkhorn-iteration.pdf>`__, + Advances in Neural Information Processing Systems (NIPS) 31 + +.. [23] Aude, G., Peyré, G., Cuturi, M., `Learning Generative Models with + Sinkhorn Divergences <https://arxiv.org/abs/1706.00292>`__, Proceedings + of the Twenty-First International Conference on Artficial Intelligence + and Statistics, (AISTATS) 21, 2018 + +.. [24] Vayer, T., Chapel, L., Flamary, R., Tavenard, R. and Courty, N. + (2019). `Optimal Transport for structured data with application on + graphs <http://proceedings.mlr.press/v97/titouan19a.html>`__ Proceedings + of the 36th International Conference on Machine Learning (ICML).
\ No newline at end of file diff --git a/docs/source/readme.rst b/docs/source/readme.rst index e7c2bd1..0871779 100644 --- a/docs/source/readme.rst +++ b/docs/source/readme.rst @@ -12,9 +12,11 @@ It provides the following solvers: - OT Network Flow solver for the linear program/ Earth Movers Distance [1]. -- Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2] - and stabilized version [9][10] and greedy SInkhorn [22] with optional - GPU implementation (requires cudamat). +- Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2], + stabilized version [9][10] and greedy Sinkhorn [22] with optional GPU + implementation (requires cupy). +- Sinkhorn divergence [23] and entropic regularization OT from + empirical data. - Smooth optimal transport solvers (dual and semi-dual) for KL and squared L2 regularizations [17]. - Non regularized Wasserstein barycenters [16] with LP solver (only @@ -33,6 +35,7 @@ It provides the following solvers: - Stochastic Optimization for Large-scale Optimal Transport (semi-dual problem [18] and dual problem [19]) - Non regularized free support Wasserstein barycenters [20]. +- Unbalanced OT with KL relaxation distance and barycenter [10, 25]. Some demonstrations (both in Python and Jupyter Notebook format) are available in the examples folder. @@ -67,6 +70,13 @@ modules: Pip installation ^^^^^^^^^^^^^^^^ +Note that due to a limitation of pip, ``cython`` and ``numpy`` need to +be installed prior to installing POT. This can be done easily with + +:: + + pip install numpy cython + You can install the toolbox through PyPI with: :: @@ -115,14 +125,9 @@ below pip install pymanopt autograd -- **ot.gpu** (GPU accelerated OT) depends on cudamat that have to be - installed with: - - :: - - git clone https://github.com/cudamat/cudamat.git - cd cudamat - python setup.py install --user # for user install (no root) +- **ot.gpu** (GPU accelerated OT) depends on cupy that have to be + installed following instructions on `this + page <https://docs-cupy.chainer.org/en/stable/install.html>`__. obviously you need CUDA installed and a compatible GPU. @@ -209,10 +214,13 @@ nbviewer <https://nbviewer.jupyter.org/github/rflamary/POT/tree/master/notebooks Acknowledgements ---------------- -The contributors to this library are: +This toolbox has been created and is maintained by - `Rémi Flamary <http://remi.flamary.com/>`__ - `Nicolas Courty <http://people.irisa.fr/Nicolas.Courty/>`__ + +The contributors to this library are + - `Alexandre Gramfort <http://alexandre.gramfort.net/>`__ - `Laetitia Chapel <http://people.irisa.fr/Laetitia.Chapel/>`__ - `Michael Perrot <http://perso.univ-st-etienne.fr/pem82055/>`__ @@ -226,6 +234,9 @@ The contributors to this library are: - `Kilian Fatras <https://kilianfatras.github.io/>`__ - `Alain Rakotomamonjy <https://sites.google.com/site/alainrakotomamonjy/home>`__ +- `Vayer Titouan <https://tvayer.github.io/>`__ +- `Hicham Janati <https://hichamjanati.github.io/>`__ (Unbalanced OT) +- `Romain Tavenard <https://rtavenar.github.io/>`__ (1d Wasserstein) This toolbox benefit a lot from open source research and we would like to thank the following persons for providing some code (in various @@ -366,6 +377,20 @@ approximation algorithms for optimal transport via Sinkhorn iteration <https://papers.nips.cc/paper/6792-near-linear-time-approximation-algorithms-for-optimal-transport-via-sinkhorn-iteration.pdf>`__, Advances in Neural Information Processing Systems (NIPS) 31 +[23] Aude, G., Peyré, G., Cuturi, M., `Learning Generative Models with +Sinkhorn Divergences <https://arxiv.org/abs/1706.00292>`__, Proceedings +of the Twenty-First International Conference on Artficial Intelligence +and Statistics, (AISTATS) 21, 2018 + +[24] Vayer, T., Chapel, L., Flamary, R., Tavenard, R. and Courty, N. +(2019). `Optimal Transport for structured data with application on +graphs <http://proceedings.mlr.press/v97/titouan19a.html>`__ Proceedings +of the 36th International Conference on Machine Learning (ICML). + +[25] Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. (2019). +`Learning with a Wasserstein Loss <http://cbcl.mit.edu/wasserstein/>`__ +Advances in Neural Information Processing Systems (NIPS). + .. |PyPI version| image:: https://badge.fury.io/py/POT.svg :target: https://badge.fury.io/py/POT .. |Anaconda Cloud| image:: https://anaconda.org/conda-forge/pot/badges/version.svg |
