245 lines
9.5 KiB
Plaintext
245 lines
9.5 KiB
Plaintext
{
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"nbformat": 4,
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"nbformat_minor": 0,
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"metadata": {
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"colab": {
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"provenance": [],
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"include_colab_link": true
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},
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"kernelspec": {
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"name": "python3",
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"display_name": "Python 3"
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},
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"language_info": {
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"name": "python"
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}
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},
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "view-in-github",
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"colab_type": "text"
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},
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"source": [
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"<a href=\"https://colab.research.google.com/github/udlbook/udlbook/blob/main/Notebooks/Chap15/15_2_Wasserstein_Distance.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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]
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},
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{
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"cell_type": "markdown",
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"source": [
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"# **Notebook 15.2: Wasserstein Distance**\n",
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"\n",
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"This notebook investigates the GAN toy example as illustrated in figure 15.1 in the book.\n",
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"\n",
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"Work through the cells below, running each cell in turn. In various places you will see the words \"TO DO\". Follow the instructions at these places and make predictions about what is going to happen or write code to complete the functions.\n",
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"\n",
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"Contact me at udlbookmail@gmail.com if you find any mistakes or have any suggestions."
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],
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"metadata": {
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"id": "t9vk9Elugvmi"
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}
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},
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{
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"cell_type": "code",
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"from matplotlib import cm\n",
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"from matplotlib.colors import ListedColormap\n",
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"from scipy.optimize import linprog"
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],
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"metadata": {
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"id": "OLComQyvCIJ7"
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},
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"execution_count": null,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"# Define two probability distributions\n",
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"p = np.array([5, 3, 2, 1, 8, 7, 5, 9, 2, 1])\n",
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"q = np.array([4, 10,1, 1, 4, 6, 3, 2, 0, 1])\n",
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"p = p/np.sum(p);\n",
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"q= q/np.sum(q);\n",
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"\n",
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"# Draw those distributions\n",
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"fig, ax =plt.subplots(2,1);\n",
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"x = np.arange(0,p.size,1)\n",
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"ax[0].bar(x,p, color=\"#cccccc\")\n",
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"ax[0].set_ylim([0,0.35])\n",
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"ax[0].set_ylabel(\"p(x=i)\")\n",
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"\n",
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"ax[1].bar(x,q,color=\"#f47a60\")\n",
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"ax[1].set_ylim([0,0.35])\n",
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"ax[1].set_ylabel(\"q(x=j)\")\n",
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"plt.show()"
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],
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"metadata": {
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"id": "ZIfQwhd-AV6L"
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},
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"execution_count": null,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"# TODO Define the distance matrix from figure 15.8d\n",
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"# Replace this line\n",
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"dist_mat = np.zeros((10,10))\n",
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"\n",
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"# vectorize the distance matrix\n",
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"c = dist_mat.flatten()"
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],
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"metadata": {
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"id": "EZSlZQzWBKTm"
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},
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"execution_count": null,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"# Define pretty colormap\n",
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"my_colormap_vals_hex =('2a0902', '2b0a03', '2c0b04', '2d0c05', '2e0c06', '2f0d07', '300d08', '310e09', '320f0a', '330f0b', '34100b', '35110c', '36110d', '37120e', '38120f', '39130f', '3a1410', '3b1411', '3c1511', '3d1612', '3e1613', '3f1713', '401714', '411814', '421915', '431915', '451a16', '461b16', '471b17', '481c17', '491d18', '4a1d18', '4b1e19', '4c1f19', '4d1f1a', '4e201b', '50211b', '51211c', '52221c', '53231d', '54231d', '55241e', '56251e', '57261f', '58261f', '592720', '5b2821', '5c2821', '5d2922', '5e2a22', '5f2b23', '602b23', '612c24', '622d25', '632e25', '652e26', '662f26', '673027', '683027', '693128', '6a3229', '6b3329', '6c342a', '6d342a', '6f352b', '70362c', '71372c', '72372d', '73382e', '74392e', '753a2f', '763a2f', '773b30', '783c31', '7a3d31', '7b3e32', '7c3e33', '7d3f33', '7e4034', '7f4134', '804235', '814236', '824336', '834437', '854538', '864638', '874739', '88473a', '89483a', '8a493b', '8b4a3c', '8c4b3c', '8d4c3d', '8e4c3e', '8f4d3f', '904e3f', '924f40', '935041', '945141', '955242', '965343', '975343', '985444', '995545', '9a5646', '9b5746', '9c5847', '9d5948', '9e5a49', '9f5a49', 'a05b4a', 'a15c4b', 'a35d4b', 'a45e4c', 'a55f4d', 'a6604e', 'a7614e', 'a8624f', 'a96350', 'aa6451', 'ab6552', 'ac6552', 'ad6653', 'ae6754', 'af6855', 'b06955', 'b16a56', 'b26b57', 'b36c58', 'b46d59', 'b56e59', 'b66f5a', 'b7705b', 'b8715c', 'b9725d', 'ba735d', 'bb745e', 'bc755f', 'bd7660', 'be7761', 'bf7862', 'c07962', 'c17a63', 'c27b64', 'c27c65', 'c37d66', 'c47e67', 'c57f68', 'c68068', 'c78169', 'c8826a', 'c9836b', 'ca846c', 'cb856d', 'cc866e', 'cd876f', 'ce886f', 'ce8970', 'cf8a71', 'd08b72', 'd18c73', 'd28d74', 'd38e75', 'd48f76', 'd59077', 'd59178', 'd69279', 'd7937a', 'd8957b', 'd9967b', 'da977c', 'da987d', 'db997e', 'dc9a7f', 'dd9b80', 'de9c81', 'de9d82', 'df9e83', 'e09f84', 'e1a185', 'e2a286', 'e2a387', 'e3a488', 'e4a589', 'e5a68a', 'e5a78b', 'e6a88c', 'e7aa8d', 'e7ab8e', 'e8ac8f', 'e9ad90', 'eaae91', 'eaaf92', 'ebb093', 'ecb295', 'ecb396', 'edb497', 'eeb598', 'eeb699', 'efb79a', 'efb99b', 'f0ba9c', 'f1bb9d', 'f1bc9e', 'f2bd9f', 'f2bfa1', 'f3c0a2', 'f3c1a3', 'f4c2a4', 'f5c3a5', 'f5c5a6', 'f6c6a7', 'f6c7a8', 'f7c8aa', 'f7c9ab', 'f8cbac', 'f8ccad', 'f8cdae', 'f9ceb0', 'f9d0b1', 'fad1b2', 'fad2b3', 'fbd3b4', 'fbd5b6', 'fbd6b7', 'fcd7b8', 'fcd8b9', 'fcdaba', 'fddbbc', 'fddcbd', 'fddebe', 'fddfbf', 'fee0c1', 'fee1c2', 'fee3c3', 'fee4c5', 'ffe5c6', 'ffe7c7', 'ffe8c9', 'ffe9ca', 'ffebcb', 'ffeccd', 'ffedce', 'ffefcf', 'fff0d1', 'fff2d2', 'fff3d3', 'fff4d5', 'fff6d6', 'fff7d8', 'fff8d9', 'fffada', 'fffbdc', 'fffcdd', 'fffedf', 'ffffe0')\n",
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"my_colormap_vals_dec = np.array([int(element,base=16) for element in my_colormap_vals_hex])\n",
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"r = np.floor(my_colormap_vals_dec/(256*256))\n",
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"g = np.floor((my_colormap_vals_dec - r *256 *256)/256)\n",
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"b = np.floor(my_colormap_vals_dec - r * 256 *256 - g * 256)\n",
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"my_colormap = ListedColormap(np.vstack((r,g,b)).transpose()/255.0)\n",
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"\n",
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"def draw_2D_heatmap(data, title, my_colormap):\n",
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" # Make grid of intercept/slope values to plot\n",
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" xv, yv = np.meshgrid(np.linspace(0, 1, 10), np.linspace(0, 1, 10))\n",
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" fig,ax = plt.subplots()\n",
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" fig.set_size_inches(4,4)\n",
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" plt.imshow(data, cmap=my_colormap)\n",
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" ax.set_title(title)\n",
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" ax.set_xlabel('$q$'); ax.set_ylabel('$p$')\n",
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" plt.show()"
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],
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"metadata": {
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"id": "ABRANmp6F8iQ"
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},
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"execution_count": null,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"draw_2D_heatmap(dist_mat,'Distance $|i-j|$', my_colormap)"
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],
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"metadata": {
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"id": "G0HFPBXyHT6V"
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},
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"execution_count": null,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"# Define b to be the verticalconcatenation of p and q\n",
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"b = np.hstack((p,q))[np.newaxis].transpose()"
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],
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"metadata": {
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"id": "SfqeT3KlHWrt"
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},
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"execution_count": null,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"# TODO: Now construct the matrix A that has the initial distribution constraints\n",
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"# so that A @ TPFlat=b where TPFlat is the transport plan TP vectorized rows first so TPFlat = np.flatten(TP)\n",
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"# Replace this line:\n",
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"A = np.zeros((20,100))"
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],
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"metadata": {
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"id": "7KrybL96IuNW"
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},
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"execution_count": null,
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"outputs": []
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},
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{
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"cell_type": "markdown",
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"source": [
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"Now we have all of the things we need. The vectorized distance matrix $\\mathbf{c}$, the constraint matrix $\\mathbf{A}$, the vectorized and concatenated original distribution $\\mathbf{b}$. We can run the linear programming optimization."
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],
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"metadata": {
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"id": "zEuEtU33S8Ly"
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}
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},
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{
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"cell_type": "code",
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"source": [
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"# We don't need the constraint that p>0 as this is the default\n",
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"opt = linprog(c, A_eq=A, b_eq=b)\n",
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"print(opt)"
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],
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"metadata": {
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"id": "wCfsOVbeSmF5"
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},
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"execution_count": null,
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"outputs": []
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},
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{
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"cell_type": "markdown",
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"source": [
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"Extract the answer and display"
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],
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"metadata": {
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"id": "vpkkOOI2agyl"
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}
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},
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{
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"cell_type": "code",
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"source": [
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"TP = np.array(opt.x).reshape(10,10)\n",
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"draw_2D_heatmap(TP,'Transport plan $\\mathbf{P}$', my_colormap)"
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],
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"metadata": {
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"id": "nZGfkrbRV_D0"
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},
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"execution_count": null,
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"outputs": []
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},
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{
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"cell_type": "markdown",
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"source": [
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"Compute the Wasserstein distance\n"
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],
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"metadata": {
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"id": "ZEiRYRVgalsJ"
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}
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},
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{
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"cell_type": "code",
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"source": [
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"was = np.sum(TP * dist_mat)\n",
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"print(\"Wasserstein distance = \", was)"
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],
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"metadata": {
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"id": "yiQ_8j-Raq3c"
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},
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"execution_count": null,
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"outputs": []
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},
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{
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"cell_type": "markdown",
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"source": [
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"TODO -- Compute the\n",
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"\n",
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"* Forward KL divergence $D_{KL}[p,q]$ between these distributions\n",
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"* Reverse KL divergence $D_{KL}[q,p]$ between these distributions\n",
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"* Jensen-Shannon divergence $D_{JS}[p,q]$ between these distributions\n",
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"\n",
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"What do you conclude?"
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],
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"metadata": {
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"id": "zf8yTusua71s"
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}
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}
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]
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} |