{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\n# Weak Optimal Transport VS exact Optimal Transport\n\n<div class=\"alert alert-info\"><h4>Note</h4><p>Example added in release: 0.8.2.</p></div>\n\nIllustration of 2D optimal transport between distributions that are weighted\nsum of Diracs. The OT matrix is plotted with the samples.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
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      "source": [
        "# Author: Remi Flamary <remi.flamary@polytechnique.edu>\n#\n# License: MIT License\n\n# sphinx_gallery_thumbnail_number = 4\n\nimport numpy as np\nimport matplotlib.pylab as pl\nimport ot\nimport ot.plot"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Generate data an plot it\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "n = 50  # nb samples\n\nmu_s = np.array([0, 0])\ncov_s = np.array([[1, 0], [0, 1]])\n\nmu_t = np.array([4, 4])\ncov_t = np.array([[1, -0.8], [-0.8, 1]])\n\nxs = ot.datasets.make_2D_samples_gauss(n, mu_s, cov_s)\nxt = ot.datasets.make_2D_samples_gauss(n, mu_t, cov_t)\n\na, b = ot.unif(n), ot.unif(n)  # uniform distribution on samples\n\n# loss matrix\nM = ot.dist(xs, xt)\nM /= M.max()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "pl.figure(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(\"Source and target distributions\")\n\npl.figure(2)\npl.imshow(M, interpolation=\"nearest\")\npl.title(\"Cost matrix M\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Compute Weak OT and exact OT solutions\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "G0 = ot.emd(a, b, M)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "Gweak = ot.weak_optimal_transport(xs, xt, a, b)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Plot weak OT and exact OT solutions\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "pl.figure(3, (8, 5))\n\npl.subplot(1, 2, 1)\npl.imshow(G0, interpolation=\"nearest\")\npl.title(\"OT matrix\")\n\npl.subplot(1, 2, 2)\npl.imshow(Gweak, interpolation=\"nearest\")\npl.title(\"Weak OT matrix\")\n\npl.figure(4, (8, 5))\n\npl.subplot(1, 2, 1)\not.plot.plot2D_samples_mat(xs, xt, G0, c=[0.5, 0.5, 1])\npl.plot(xs[:, 0], xs[:, 1], \"+b\", label=\"Source samples\")\npl.plot(xt[:, 0], xt[:, 1], \"xr\", label=\"Target samples\")\npl.title(\"OT matrix with samples\")\n\npl.subplot(1, 2, 2)\not.plot.plot2D_samples_mat(xs, xt, Gweak, c=[0.5, 0.5, 1])\npl.plot(xs[:, 0], xs[:, 1], \"+b\", label=\"Source samples\")\npl.plot(xt[:, 0], xt[:, 1], \"xr\", label=\"Target samples\")\npl.title(\"Weak OT matrix with samples\")"
      ]
    }
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