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diff --git a/docs/source/auto_examples/plot_barycenter_lp_vs_entropic.rst b/docs/source/auto_examples/plot_barycenter_lp_vs_entropic.rst deleted file mode 100644 index bd1c710..0000000 --- a/docs/source/auto_examples/plot_barycenter_lp_vs_entropic.rst +++ /dev/null @@ -1,447 +0,0 @@ - - -.. _sphx_glr_auto_examples_plot_barycenter_lp_vs_entropic.py: - - -================================================================================= -1D Wasserstein barycenter comparison between exact LP and entropic regularization -================================================================================= - -This example illustrates the computation of regularized Wasserstein Barycenter -as proposed in [3] and exact LP barycenters using standard LP solver. - -It reproduces approximately Figure 3.1 and 3.2 from the following paper: -Cuturi, M., & Peyré, G. (2016). A smoothed dual approach for variational -Wasserstein problems. SIAM Journal on Imaging Sciences, 9(1), 320-343. - -[3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015). -Iterative Bregman projections for regularized transportation problems -SIAM Journal on Scientific Computing, 37(2), A1111-A1138. - - - - -.. code-block:: python - - - # Author: Remi Flamary <remi.flamary@unice.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 # noqa - - #import ot.lp.cvx as cvx - - - - - - - -Gaussian Data -------------- - - - -.. code-block:: python - - - #%% parameters - - problems = [] - - n = 100 # nb bins - - # bin positions - x = np.arange(n, dtype=np.float64) - - # Gaussian distributions - # 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) - - # 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 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 - - alpha = 0.5 # 0<=alpha<=1 - weights = np.array([1 - alpha, alpha]) - - # l2bary - bary_l2 = A.dot(weights) - - # wasserstein - reg = 1e-3 - ot.tic() - bary_wass = ot.bregman.barycenter(A, M, reg, weights) - ot.toc() - - - ot.tic() - bary_wass2 = ot.lp.barycenter(A, M, weights, solver='interior-point', verbose=True) - ot.toc() - - 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='Reg Wasserstein') - pl.plot(x, bary_wass2, 'b', label='LP Wasserstein') - pl.legend() - pl.title('Barycenters') - pl.tight_layout() - - problems.append([A, [bary_l2, bary_wass, bary_wass2]]) - - - - -.. rst-class:: sphx-glr-horizontal - - - * - - .. image:: /auto_examples/images/sphx_glr_plot_barycenter_lp_vs_entropic_001.png - :scale: 47 - - * - - .. image:: /auto_examples/images/sphx_glr_plot_barycenter_lp_vs_entropic_002.png - :scale: 47 - - -.. rst-class:: sphx-glr-script-out - - Out:: - - Elapsed time : 0.010712385177612305 s - Primal Feasibility Dual Feasibility Duality Gap Step Path Parameter Objective - 1.0 1.0 1.0 - 1.0 1700.336700337 - 0.006776453137632 0.006776453137633 0.006776453137633 0.9932238647293 0.006776453137633 125.6700527543 - 0.004018712867874 0.004018712867874 0.004018712867874 0.4301142633 0.004018712867874 12.26594150093 - 0.001172775061627 0.001172775061627 0.001172775061627 0.7599932455029 0.001172775061627 0.3378536968897 - 0.0004375137005385 0.0004375137005385 0.0004375137005385 0.6422331807989 0.0004375137005385 0.1468420566358 - 0.000232669046734 0.0002326690467341 0.000232669046734 0.5016999460893 0.000232669046734 0.09381703231432 - 7.430121674303e-05 7.430121674303e-05 7.430121674303e-05 0.7035962305812 7.430121674303e-05 0.0577787025717 - 5.321227838876e-05 5.321227838875e-05 5.321227838876e-05 0.308784186441 5.321227838876e-05 0.05266249477203 - 1.990900379199e-05 1.990900379196e-05 1.990900379199e-05 0.6520472013244 1.990900379199e-05 0.04526054405519 - 6.305442046799e-06 6.30544204682e-06 6.3054420468e-06 0.7073953304075 6.305442046798e-06 0.04237597591383 - 2.290148391577e-06 2.290148391582e-06 2.290148391578e-06 0.6941812711492 2.29014839159e-06 0.041522849321 - 1.182864875387e-06 1.182864875406e-06 1.182864875427e-06 0.508455204675 1.182864875445e-06 0.04129461872827 - 3.626786381529e-07 3.626786382468e-07 3.626786382923e-07 0.7101651572101 3.62678638267e-07 0.04113032448923 - 1.539754244902e-07 1.539754249276e-07 1.539754249356e-07 0.6279322066282 1.539754253892e-07 0.04108867636379 - 5.193221323143e-08 5.193221463044e-08 5.193221462729e-08 0.6843453436759 5.193221708199e-08 0.04106859618414 - 1.888205054507e-08 1.888204779723e-08 1.88820477688e-08 0.6673444085651 1.888205650952e-08 0.041062141752 - 5.676855206925e-09 5.676854518888e-09 5.676854517651e-09 0.7281705804232 5.676885442702e-09 0.04105958648713 - 3.501157668218e-09 3.501150243546e-09 3.501150216347e-09 0.414020345194 3.501164437194e-09 0.04105916265261 - 1.110594251499e-09 1.110590786827e-09 1.11059083379e-09 0.6998954759911 1.110636623476e-09 0.04105870073485 - 5.770971626386e-10 5.772456113791e-10 5.772456200156e-10 0.4999769658132 5.77013379477e-10 0.04105859769135 - 1.535218204536e-10 1.536993317032e-10 1.536992771966e-10 0.7516471627141 1.536205005991e-10 0.04105851679958 - 6.724209350756e-11 6.739211232927e-11 6.739210470901e-11 0.5944802416166 6.735465384341e-11 0.04105850033766 - 1.743382199199e-11 1.736445896691e-11 1.736448490761e-11 0.7573407808104 1.734254328931e-11 0.04105849088824 - Optimization terminated successfully. - Elapsed time : 2.883899211883545 s - - -Dirac Data ----------- - - - -.. code-block:: python - - - #%% parameters - - a1 = 1.0 * (x > 10) * (x < 50) - a2 = 1.0 * (x > 60) * (x < 80) - - a1 /= a1.sum() - a2 /= a2.sum() - - # 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 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 - - alpha = 0.5 # 0<=alpha<=1 - weights = np.array([1 - alpha, alpha]) - - # l2bary - bary_l2 = A.dot(weights) - - # wasserstein - reg = 1e-3 - ot.tic() - bary_wass = ot.bregman.barycenter(A, M, reg, weights) - ot.toc() - - - ot.tic() - bary_wass2 = ot.lp.barycenter(A, M, weights, solver='interior-point', verbose=True) - ot.toc() - - - problems.append([A, [bary_l2, bary_wass, bary_wass2]]) - - 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='Reg Wasserstein') - pl.plot(x, bary_wass2, 'b', label='LP Wasserstein') - pl.legend() - pl.title('Barycenters') - pl.tight_layout() - - #%% parameters - - a1 = np.zeros(n) - a2 = np.zeros(n) - - a1[10] = .25 - a1[20] = .5 - a1[30] = .25 - a2[80] = 1 - - - a1 /= a1.sum() - a2 /= a2.sum() - - # 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 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 - - alpha = 0.5 # 0<=alpha<=1 - weights = np.array([1 - alpha, alpha]) - - # l2bary - bary_l2 = A.dot(weights) - - # wasserstein - reg = 1e-3 - ot.tic() - bary_wass = ot.bregman.barycenter(A, M, reg, weights) - ot.toc() - - - ot.tic() - bary_wass2 = ot.lp.barycenter(A, M, weights, solver='interior-point', verbose=True) - ot.toc() - - - problems.append([A, [bary_l2, bary_wass, bary_wass2]]) - - 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='Reg Wasserstein') - pl.plot(x, bary_wass2, 'b', label='LP Wasserstein') - pl.legend() - pl.title('Barycenters') - pl.tight_layout() - - - - - -.. rst-class:: sphx-glr-horizontal - - - * - - .. image:: /auto_examples/images/sphx_glr_plot_barycenter_lp_vs_entropic_003.png - :scale: 47 - - * - - .. image:: /auto_examples/images/sphx_glr_plot_barycenter_lp_vs_entropic_004.png - :scale: 47 - - -.. rst-class:: sphx-glr-script-out - - Out:: - - Elapsed time : 0.014938592910766602 s - Primal Feasibility Dual Feasibility Duality Gap Step Path Parameter Objective - 1.0 1.0 1.0 - 1.0 1700.336700337 - 0.006776466288966 0.006776466288966 0.006776466288966 0.9932238515788 0.006776466288966 125.6649255808 - 0.004036918865495 0.004036918865495 0.004036918865495 0.4272973099316 0.004036918865495 12.3471617011 - 0.00121923268707 0.00121923268707 0.00121923268707 0.749698685599 0.00121923268707 0.3243835647408 - 0.0003837422984432 0.0003837422984432 0.0003837422984432 0.6926882608284 0.0003837422984432 0.1361719397493 - 0.0001070128410183 0.0001070128410183 0.0001070128410183 0.7643889137854 0.0001070128410183 0.07581952832518 - 0.0001001275033711 0.0001001275033711 0.0001001275033711 0.07058704837812 0.0001001275033712 0.0734739493635 - 4.550897507844e-05 4.550897507841e-05 4.550897507844e-05 0.5761172484828 4.550897507845e-05 0.05555077655047 - 8.557124125522e-06 8.5571241255e-06 8.557124125522e-06 0.8535925441152 8.557124125522e-06 0.04439814660221 - 3.611995628407e-06 3.61199562841e-06 3.611995628414e-06 0.6002277331554 3.611995628415e-06 0.04283007762152 - 7.590393750365e-07 7.590393750491e-07 7.590393750378e-07 0.8221486533416 7.590393750381e-07 0.04192322976248 - 8.299929287441e-08 8.299929286079e-08 8.299929287532e-08 0.9017467938799 8.29992928758e-08 0.04170825633295 - 3.117560203449e-10 3.117560130137e-10 3.11756019954e-10 0.997039969226 3.11756019952e-10 0.04168179329766 - 1.559749653711e-14 1.558073160926e-14 1.559756940692e-14 0.9999499686183 1.559750643989e-14 0.04168169240444 - Optimization terminated successfully. - Elapsed time : 2.642659902572632 s - Elapsed time : 0.002908945083618164 s - Primal Feasibility Dual Feasibility Duality Gap Step Path Parameter Objective - 1.0 1.0 1.0 - 1.0 1700.336700337 - 0.006774675520727 0.006774675520727 0.006774675520727 0.9932256422636 0.006774675520727 125.6956034743 - 0.002048208707562 0.002048208707562 0.002048208707562 0.7343095368143 0.002048208707562 5.213991622123 - 0.000269736547478 0.0002697365474781 0.0002697365474781 0.8839403501193 0.000269736547478 0.505938390389 - 6.832109993943e-05 6.832109993944e-05 6.832109993944e-05 0.7601171075965 6.832109993943e-05 0.2339657807272 - 2.437682932219e-05 2.43768293222e-05 2.437682932219e-05 0.6663448297475 2.437682932219e-05 0.1471256246325 - 1.13498321631e-05 1.134983216308e-05 1.13498321631e-05 0.5553643816404 1.13498321631e-05 0.1181584941171 - 3.342312725885e-06 3.342312725884e-06 3.342312725885e-06 0.7238133571615 3.342312725885e-06 0.1006387519747 - 7.078561231603e-07 7.078561231509e-07 7.078561231604e-07 0.8033142552512 7.078561231603e-07 0.09474734646269 - 1.966870956916e-07 1.966870954537e-07 1.966870954468e-07 0.752547917788 1.966870954633e-07 0.09354342735766 - 4.19989524849e-10 4.199895164852e-10 4.199895238758e-10 0.9984019849375 4.19989523951e-10 0.09310367785861 - 2.101015938666e-14 2.100625691113e-14 2.101023853438e-14 0.999949974425 2.101023691864e-14 0.09310274466458 - Optimization terminated successfully. - Elapsed time : 2.690450668334961 s - - -Final figure ------------- - - - - -.. code-block:: python - - - #%% plot - - nbm = len(problems) - nbm2 = (nbm // 2) - - - pl.figure(2, (20, 6)) - pl.clf() - - for i in range(nbm): - - A = problems[i][0] - bary_l2 = problems[i][1][0] - bary_wass = problems[i][1][1] - bary_wass2 = problems[i][1][2] - - pl.subplot(2, nbm, 1 + i) - for j in range(n_distributions): - pl.plot(x, A[:, j]) - if i == nbm2: - pl.title('Distributions') - pl.xticks(()) - pl.yticks(()) - - pl.subplot(2, nbm, 1 + i + nbm) - - pl.plot(x, bary_l2, 'r', label='L2 (Euclidean)') - pl.plot(x, bary_wass, 'g', label='Reg Wasserstein') - pl.plot(x, bary_wass2, 'b', label='LP Wasserstein') - if i == nbm - 1: - pl.legend() - if i == nbm2: - pl.title('Barycenters') - - pl.xticks(()) - pl.yticks(()) - - - -.. image:: /auto_examples/images/sphx_glr_plot_barycenter_lp_vs_entropic_006.png - :align: center - - - - -**Total running time of the script:** ( 0 minutes 8.892 seconds) - - - -.. only :: html - - .. container:: sphx-glr-footer - - - .. container:: sphx-glr-download - - :download:`Download Python source code: plot_barycenter_lp_vs_entropic.py <plot_barycenter_lp_vs_entropic.py>` - - - - .. container:: sphx-glr-download - - :download:`Download Jupyter notebook: plot_barycenter_lp_vs_entropic.ipynb <plot_barycenter_lp_vs_entropic.ipynb>` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_ |