{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\n# Optimal Transport for fixed support\n\nThis example illustrates the computation of EMD and Sinkhorn transport plans\nand their visualization.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "# Author: Remi Flamary <remi.flamary@unice.fr>\n#\n# License: MIT License\n# sphinx_gallery_thumbnail_number = 3\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"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "n = 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# 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"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "pl.figure(1, figsize=(6.4, 3))\npl.plot(x, a, \"b\", label=\"Source distribution\")\npl.plot(x, b, \"r\", label=\"Target distribution\")\npl.legend()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "pl.figure(2, figsize=(5, 5))\not.plot.plot1D_mat(a, b, M, \"Cost matrix M\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Solve Exact OT\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "# use fast 1D solver\nG0 = ot.emd_1d(x, x, a, b)\n\n# Equivalent to\n# G0 = ot.emd(a, b, M)\n\npl.figure(3, figsize=(5, 5))\not.plot.plot1D_mat(a, b, G0, \"OT matrix G0\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Solve Sinkhorn\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "lambd = 1e-3\nGs = ot.sinkhorn(a, b, M, lambd, verbose=True)\n\npl.figure(4, figsize=(5, 5))\not.plot.plot1D_mat(a, b, Gs, \"OT matrix Sinkhorn\")\n\npl.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Solve Smooth OT\nWe illustrate below Smooth and Sparse (KL an L2 reg.) OT and\nsparsity-constrained OT, together with their visualizations.\n\nReference:\n\nBlondel, M., Seguy, V., & Rolet, A. (2018). Smooth and Sparse Optimal\nTransport. Proceedings of the # Twenty-First International Conference on\nArtificial Intelligence and # Statistics (AISTATS).\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "lambd = 2e-3\nGsm = ot.smooth.smooth_ot_dual(a, b, M, lambd, reg_type=\"kl\")\n\npl.figure(3, figsize=(5, 5))\not.plot.plot1D_mat(a, b, Gsm, \"OT matrix Smooth OT KL reg.\")\n\npl.show()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "lambd = 1e-1\nGsm = ot.smooth.smooth_ot_dual(a, b, M, lambd, reg_type=\"l2\")\n\npl.figure(5, figsize=(5, 5))\not.plot.plot1D_mat(a, b, Gsm, \"OT matrix Smooth OT l2 reg.\")\n\npl.show()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "lambd = 1e-1\n\nmax_nz = 2  # two non-zero entries are permitted per column of the OT plan\nGsc = ot.smooth.smooth_ot_dual(\n    a, b, M, lambd, reg_type=\"sparsity_constrained\", max_nz=max_nz\n)\npl.figure(6, figsize=(5, 5))\not.plot.plot1D_mat(a, b, Gsc, \"Sparsity constrained OT matrix; k=2.\")\n\npl.show()"
      ]
    }
  ],
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