.. _sphx_glr_auto_examples_plot_OT_L1_vs_L2.py: ========================================== 2D Optimal transport for different metrics ========================================== 2D OT on empirical distributio with different gound metric. Stole the figure idea from Fig. 1 and 2 in https://arxiv.org/pdf/1706.07650.pdf .. code-block:: python # Author: Remi Flamary # # License: MIT License import numpy as np import matplotlib.pylab as pl import ot import ot.plot Dataset 1 : uniform sampling ---------------------------- .. code-block:: python n = 20 # nb samples xs = np.zeros((n, 2)) xs[:, 0] = np.arange(n) + 1 xs[:, 1] = (np.arange(n) + 1) * -0.001 # to make it strictly convex... xt = np.zeros((n, 2)) xt[:, 1] = np.arange(n) + 1 a, b = ot.unif(n), ot.unif(n) # uniform distribution on samples # loss matrix M1 = ot.dist(xs, xt, metric='euclidean') M1 /= M1.max() # loss matrix M2 = ot.dist(xs, xt, metric='sqeuclidean') M2 /= M2.max() # loss matrix Mp = np.sqrt(ot.dist(xs, xt, metric='euclidean')) Mp /= Mp.max() # Data pl.figure(1, figsize=(7, 3)) pl.clf() pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples') pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples') pl.axis('equal') pl.title('Source and target distributions') # Cost matrices pl.figure(2, figsize=(7, 3)) pl.subplot(1, 3, 1) pl.imshow(M1, interpolation='nearest') pl.title('Euclidean cost') pl.subplot(1, 3, 2) pl.imshow(M2, interpolation='nearest') pl.title('Squared Euclidean cost') pl.subplot(1, 3, 3) pl.imshow(Mp, interpolation='nearest') pl.title('Sqrt Euclidean cost') pl.tight_layout() .. rst-class:: sphx-glr-horizontal * .. image:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_001.png :scale: 47 * .. image:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_002.png :scale: 47 Dataset 1 : Plot OT Matrices ---------------------------- .. code-block:: python #%% EMD G1 = ot.emd(a, b, M1) G2 = ot.emd(a, b, M2) Gp = ot.emd(a, b, Mp) # OT matrices pl.figure(3, figsize=(7, 3)) pl.subplot(1, 3, 1) ot.plot.plot2D_samples_mat(xs, xt, G1, c=[.5, .5, 1]) pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples') pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples') pl.axis('equal') # pl.legend(loc=0) pl.title('OT Euclidean') pl.subplot(1, 3, 2) ot.plot.plot2D_samples_mat(xs, xt, G2, c=[.5, .5, 1]) pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples') pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples') pl.axis('equal') # pl.legend(loc=0) pl.title('OT squared Euclidean') pl.subplot(1, 3, 3) ot.plot.plot2D_samples_mat(xs, xt, Gp, c=[.5, .5, 1]) pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples') pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples') pl.axis('equal') # pl.legend(loc=0) pl.title('OT sqrt Euclidean') pl.tight_layout() pl.show() .. image:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_005.png :align: center Dataset 2 : Partial circle -------------------------- .. code-block:: python n = 50 # nb samples xtot = np.zeros((n + 1, 2)) xtot[:, 0] = np.cos( (np.arange(n + 1) + 1.0) * 0.9 / (n + 2) * 2 * np.pi) xtot[:, 1] = np.sin( (np.arange(n + 1) + 1.0) * 0.9 / (n + 2) * 2 * np.pi) xs = xtot[:n, :] xt = xtot[1:, :] a, b = ot.unif(n), ot.unif(n) # uniform distribution on samples # loss matrix M1 = ot.dist(xs, xt, metric='euclidean') M1 /= M1.max() # loss matrix M2 = ot.dist(xs, xt, metric='sqeuclidean') M2 /= M2.max() # loss matrix Mp = np.sqrt(ot.dist(xs, xt, metric='euclidean')) Mp /= Mp.max() # Data pl.figure(4, figsize=(7, 3)) pl.clf() pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples') pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples') pl.axis('equal') pl.title('Source and traget distributions') # Cost matrices pl.figure(5, figsize=(7, 3)) pl.subplot(1, 3, 1) pl.imshow(M1, interpolation='nearest') pl.title('Euclidean cost') pl.subplot(1, 3, 2) pl.imshow(M2, interpolation='nearest') pl.title('Squared Euclidean cost') pl.subplot(1, 3, 3) pl.imshow(Mp, interpolation='nearest') pl.title('Sqrt Euclidean cost') pl.tight_layout() .. rst-class:: sphx-glr-horizontal * .. image:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_007.png :scale: 47 * .. image:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_008.png :scale: 47 Dataset 2 : Plot OT Matrices ----------------------------- .. code-block:: python #%% EMD G1 = ot.emd(a, b, M1) G2 = ot.emd(a, b, M2) Gp = ot.emd(a, b, Mp) # OT matrices pl.figure(6, figsize=(7, 3)) pl.subplot(1, 3, 1) ot.plot.plot2D_samples_mat(xs, xt, G1, c=[.5, .5, 1]) pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples') pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples') pl.axis('equal') # pl.legend(loc=0) pl.title('OT Euclidean') pl.subplot(1, 3, 2) ot.plot.plot2D_samples_mat(xs, xt, G2, c=[.5, .5, 1]) pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples') pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples') pl.axis('equal') # pl.legend(loc=0) pl.title('OT squared Euclidean') pl.subplot(1, 3, 3) ot.plot.plot2D_samples_mat(xs, xt, Gp, c=[.5, .5, 1]) pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples') pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples') pl.axis('equal') # pl.legend(loc=0) pl.title('OT sqrt Euclidean') pl.tight_layout() pl.show() .. image:: /auto_examples/images/sphx_glr_plot_OT_L1_vs_L2_011.png :align: center **Total running time of the script:** ( 0 minutes 0.958 seconds) .. only :: html .. container:: sphx-glr-footer .. container:: sphx-glr-download :download:`Download Python source code: plot_OT_L1_vs_L2.py ` .. container:: sphx-glr-download :download:`Download Jupyter notebook: plot_OT_L1_vs_L2.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_