Tuto-GUDHI-optimization.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Optimization and Topological Data Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mathieu Carrière, https://mathieucarriere.github.io/website/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook, we are going to see how to combine Gudhi and Tensorflow in order to do persistence optimization!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import gudhi as gd\n",
    "import tensorflow as tf\n",
    "import tensorflow_addons as tfa\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Point cloud optimization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You might have already seen a few examples of persistence diagram computations on point clouds. Among the different possibilities, the Rips filtration is the most common option due to its simplicity and easy implementation. In this notebook, we will see how Gudhi and Tensorflow can be combined to perform optimization of persistence diagrams."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's have a quick refresher ;-)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, let's generate a random point cloud in the unit square and visualize it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = np.random.uniform(size=[300,2])\n",
    "plt.scatter(X[:,0], X[:,1], s=3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Yep, looks pretty random indeed. Let's now compute its Rips persistence diagram. This is literally two lines of code with Gudhi :-)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "st = gd.RipsComplex(points=X, max_edge_length=1.).create_simplex_tree(max_dimension=2)\n",
    "dgm = st.persistence()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot = gd.plot_persistence_diagram(dgm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As usual, there is one point (in dimension 0) at $+\\infty$ which represents the whole connected component, a bunch of points in dimension 0 with abscissa 0 (actually, as many as there are points in the initial point cloud), and some points in dimension 1. There is nothing else because we only kept the 2-skeleton of the Rips complex in the previous cell."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is great. But have you ever thought about the inverse problem? That is, can you tweak the point cloud so that the corresponding persistence diagram satisfies some properties? That sounds hard. Turns out it is not if you combine Gudhi and Tensorflow ;-)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before jumping to the code, let's think about what's happening here. If you think about how persistence is computed, the coordinates of any point $p$ in a persistence diagram are actually given by the filtration values of two very specific simplices: the so-called positive and negative simplices of $p$, denoted by $\\sigma_+(p)$ and $\\sigma_-(p)$ (check chapter VII.1 in [this reference book](https://books.google.com/books/about/Computational_Topology.html?id=MDXa6gFRZuIC) for more details if you feel lost). So, we have: $$p=(f(\\sigma_+(p)), f(\\sigma_-(p))),$$\n",
    "where $f$ is the filtration function. This means that if $f$ is actually defined by some parameters $f = f_\\theta$, then the gradient $\\nabla_\\theta p$ is actually given by $\\nabla_\\theta f_\\theta(\\sigma_+(p))$ and $\\nabla_\\theta f_\\theta(\\sigma_-(p))$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Interesting, but how does this look like for the Rips filtration? Well, first, the parameters $\\theta$ are now the positions of the points in the point cloud. Second, as you may recall, the filtration value of any simplex is simply the maximal distance between any two vertices in the simplex: $f(\\{v_0,\\cdots, v_n\\})=\\|v_a-v_b\\|$, $0\\leq a,b\\leq n$, where $\\|v_a-v_b\\|\\geq \\|v_i-v_j\\|$, $\\forall 0\\leq i,j \\leq n$. This has two consequences. First, this means that one can create the persistence diagram by simply picking entries of the distance matrix between the points. Second, this also means that the gradient of $f$ only depends on the positions of $v_a$ and $v_b$: $\\nabla f=\\frac{v_a-v_b}{\\|v_a-v_b\\|}$. You can check [this article](https://arxiv.org/abs/1506.03147) for more details."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All right! So the only thing that remains to do is to compute the positive and negative simplices, right? Turns out that Gudhi can do that with the `persistence_pairs()` function. Well, let's go then!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, let's write a function that computes the positive and negative simplices associated to the persistence pairs of a Rips persistence diagram, and outputs the vertices $v_a$ and $v_b$ associated to these simplices. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Rips(DX, mel, dim, card):\n",
    "    # Parameters: DX (distance matrix), \n",
    "    #             mel (maximum edge length for Rips filtration), \n",
    "    #             dim (homological dimension), \n",
    "    #             card (number of persistence diagram points, sorted by distance-to-diagonal)\n",
    "\n",
    "    # Compute the persistence pairs with Gudhi\n",
    "    rc = gd.RipsComplex(distance_matrix=DX, max_edge_length=mel)\n",
    "    st = rc.create_simplex_tree(max_dimension=dim+1)\n",
    "    dgm = st.persistence()\n",
    "    pairs = st.persistence_pairs()\n",
    "\n",
    "    # Retrieve vertices v_a and v_b by picking the ones achieving the maximal\n",
    "    # distance among all pairwise distances between the simplex vertices\n",
    "    indices, pers = [], []\n",
    "    for s1, s2 in pairs:\n",
    "        if len(s1) == dim+1:\n",
    "            l1, l2 = np.array(s1), np.array(s2)\n",
    "            i1 = [s1[v] for v in np.unravel_index(np.argmax(DX[l1,:][:,l1]),[len(s1), len(s1)])]\n",
    "            i2 = [s2[v] for v in np.unravel_index(np.argmax(DX[l2,:][:,l2]),[len(s2), len(s2)])]\n",
    "            indices += i1\n",
    "            indices += i2\n",
    "            pers.append(st.filtration(s2) - st.filtration(s1))\n",
    "    \n",
    "    # Sort points with distance-to-diagonal\n",
    "    perm = np.argsort(pers)\n",
    "    indices = list(np.reshape(indices, [-1,4])[perm][::-1,:].flatten())\n",
    "    \n",
    "    # Output indices\n",
    "    indices = indices[:4*card] + [0 for _ in range(0,max(0,4*card-len(indices)))]\n",
    "    return list(np.array(indices, dtype=np.int32))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Second, we define a Tensorflow model whose parameters are the point coordinates, and which outputs the corresponding Rips persistence diagram."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "class RipsModel(tf.keras.Model):\n",
    "    def __init__(self, X, mel=12, dim=1, card=50):\n",
    "        super(RipsModel, self).__init__()\n",
    "        self.X = X\n",
    "        self.mel = mel\n",
    "        self.dim = dim\n",
    "        self.card = card\n",
    "        \n",
    "    def call(self):\n",
    "        m, d, c = self.mel, self.dim, self.card\n",
    "        \n",
    "        # Compute distance matrix\n",
    "        DX = tfa.losses.metric_learning.pairwise_distance(self.X)\n",
    "        DXX = tf.reshape(DX, [1, DX.shape[0], DX.shape[1]])\n",
    "        \n",
    "        # Turn numpy function into tensorflow function\n",
    "        RipsTF = lambda DX: tf.numpy_function(Rips, [DX, m, d, c], [tf.int32 for _ in range(4*c)])\n",
    "        \n",
    "        # Compute vertices associated to positive and negative simplices \n",
    "        # Don't compute gradient for this operation\n",
    "        ids = tf.nest.map_structure(tf.stop_gradient, tf.map_fn(RipsTF,DXX,dtype=[tf.int32 for _ in range(4*c)]))\n",
    "        \n",
    "        # Get persistence diagram by simply picking the corresponding entries in the distance matrix\n",
    "        dgm = tf.reshape(tf.gather_nd(DX, tf.reshape(ids, [2*c,2])), [c,2])\n",
    "        return dgm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we are done! Let's define our parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_pts    = 300   # number of points in the point clouds\n",
    "card     = 50    # max number of points in the diagrams\n",
    "hom      = 1     # homological dimension\n",
    "ml       = 12.   # max distance in Rips\n",
    "n_epochs = 30    # number of optimization steps"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We randomly initalize our point cloud."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "Xinit = np.array(np.random.uniform(size=(n_pts,2)), dtype=np.float32)\n",
    "X = tf.Variable(initial_value=Xinit, trainable=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define our Tensorflow model + optimizer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "model = RipsModel(X=X, mel=ml, dim=hom, card=card)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "optimizer = tf.keras.optimizers.Adam(learning_rate=1e-2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we can launch the optimization! Our loss is the opposite of the sum of squares of the distances to the diagonal of the points in the diagram. This will force the persistence diagram to have many prominent points, and thus the point cloud will contain as much loops as possible. Let's train  and visualize the point cloud every 10 iterations, or epochs!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for epoch in range(n_epochs+1):\n",
    "    \n",
    "    with tf.GradientTape() as tape:\n",
    "        \n",
    "        # Compute persistence diagram\n",
    "        dgm = model.call()\n",
    "        \n",
    "        # Loss is sum of squares of distances to the diagonal\n",
    "        loss = -tf.math.reduce_sum(tf.square(.5*(dgm[:,1]-dgm[:,0])))\n",
    "        \n",
    "    # Compute and apply gradients\n",
    "    gradients = tape.gradient(loss, model.trainable_variables)\n",
    "    optimizer.apply_gradients(zip(gradients, model.trainable_variables))\n",
    "\n",
    "    if epoch % 10 == 0:\n",
    "        plt.figure()\n",
    "        plt.scatter(model.X.numpy()[:,0], model.X.numpy()[:,1])\n",
    "        plt.title(\"Point cloud at epoch \" + str(epoch))\n",
    "        plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looks like there are more cycles indeed ;-)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Image optimization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can actually play the same game with images! Indeed, Gudhi contains code for computing [cubical persistence](http://www2.im.uj.edu.pl/mpd/publications/Wagner_persistence.pdf), which is very well-suited for handling images. For instance, it can be used to filter a 2D image with its pixel values. Overall, the optimization follows the exact same steps as before, except that we use pixel filtration instead of Rips filtration. This means that the parameters $\\theta$ that we are now going to optimize are the pixel values themselves, and that the gradients for positive simplex $\\nabla_\\theta f_\\theta(\\sigma_+(p))$ and negative simplex $\\nabla_\\theta f_\\theta(\\sigma_-(p))$ now simply equal $1$ for the pixels associated to $\\sigma_+(p)$ and $\\sigma_-(p)$ and $0$ for all other pixels."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fortunately, Gudhi contains a function `cofaces_of_persistence_pairs()` that exactly retrieves the pixels (or, more formally, the cofaces) of the positive and negative simplices of a persistence point $p$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's start with a function that computes those pixels."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Cubical(X, dim, card):\n",
    "    # Parameters: X (image),\n",
    "    #             dim (homological dimension), \n",
    "    #             card (number of persistence diagram points, sorted by distance-to-diagonal)\n",
    "\n",
    "    # Compute the persistence pairs with Gudhi\n",
    "    cc = gd.CubicalComplex(dimensions=X.shape, top_dimensional_cells=X.flatten())\n",
    "    cc.persistence()\n",
    "    cof = cc.cofaces_of_persistence_pairs()[0][dim]\n",
    "    \n",
    "    # Sort points with distance-to-diagonal\n",
    "    Xs = X.shape\n",
    "    pers = [X[np.unravel_index(cof[idx,1], Xs)] - X[np.unravel_index(cof[idx,0], Xs)] for idx in range(len(cof))]\n",
    "    perm = np.argsort(pers)\n",
    "    cof = cof[perm]\n",
    "    \n",
    "    # Retrieve and ouput image indices/pixels corresponding to positive and negative simplices\n",
    "    D = len(X.shape)\n",
    "    ocof = np.array([0 for _ in range(D*card*2)])\n",
    "    for idx in range(len(cof[:min(card, cof.shape[0])])):\n",
    "        ocof[D*idx:D*(idx+1)]     = np.unravel_index(cof[idx,0], X.shape)\n",
    "        ocof[D*(idx+1):D*(idx+2)] = np.unravel_index(cof[idx,1], X.shape)\n",
    "    return list(np.array(ocof.flatten(), dtype=np.int32))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As before, we now define a corresponding Tensorflow model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "class CubicalModel(tf.keras.Model):\n",
    "    def __init__(self, X, dim=1, card=50):\n",
    "        super(CubicalModel, self).__init__()\n",
    "        self.X = X\n",
    "        self.dim = dim\n",
    "        self.card = card\n",
    "        \n",
    "    def call(self):\n",
    "        d, c, D = self.dim, self.card, len(self.X.shape)\n",
    "        XX = tf.reshape(self.X, [1, self.X.shape[0], self.X.shape[1]])\n",
    "        \n",
    "        # Turn numpy function into tensorflow function\n",
    "        CbTF = lambda X: tf.numpy_function(Cubical, [X, d, c], [tf.int32 for _ in range(2*D*c)])\n",
    "        \n",
    "        # Compute pixels associated to positive and negative simplices \n",
    "        # Don't compute gradient for this operation\n",
    "        inds = tf.nest.map_structure(tf.stop_gradient, tf.map_fn(CbTF,XX,dtype=[tf.int32 for _ in range(2*D*c)]))\n",
    "        \n",
    "        # Get persistence diagram by simply picking the corresponding entries in the image\n",
    "        dgm = tf.reshape(tf.gather_nd(self.X, tf.reshape(inds, [-1,D])), [-1,2])\n",
    "        return dgm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All right! Let's apply all of these now! We are going to use the following image:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "I = np.array(pd.read_csv(\"datasets/mnist_test.csv\", header=None, sep=\",\"), dtype=np.float32)\n",
    "idx = np.argwhere(I[:,0] == 8)\n",
    "image = np.reshape(-I[idx[8],1:], [28,28])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.imshow(image)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see, the upper loop of this 8 is not complete. Since it corresponds to 1-dimensional topology, we can fix this by optimizing the pixel values so that the points in the 1-dimensional persistence diagram have maximal persistence. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We first define the network parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "card     = 50   # max number of points in the diagrams\n",
    "hom      = 1    # homological dimension\n",
    "n_epochs = 100  # number of optimization steps"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define our Tensorflow model + optimizer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = tf.Variable(initial_value=np.array(image, dtype=np.float32), trainable=True)\n",
    "model = CubicalModel(X, dim=hom, card=card)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "optimizer = tf.keras.optimizers.SGD(learning_rate=5*1e-2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This time, our loss is the sum of the squared birth coodinates of the persistence diagram points, so that the loops will appear as soon as possible, and the values of the corresponding pixels will be as dark as possible. Let's train  and visualize the image every 10 iterations, or epochs!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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OBEbAgcAIOBAYAQcCI+BAYAQcCIyAA4ERcCCwoQ8fbGbnJL3cMuk6Sa8PdaW9o229KWvbytouafBtu9Hdd7dPHHrA163QbLHTOMZlQNt6U9a2lbVd0ua1jV10IDACDgQ2ioDPdn/LyNC23pS1bWVtl7RJbdv0Y3AAm4dddCAwAg4EtqkBN7MZM5s2syObud5uzOx8evb7sVG3pSltp/m2aaXYfgVtG+k2NLNK2j4zrW0owzbLtG3o22zTAt4c4ig9a/3CwAklUbrRUtN2uqBM26+9bcmot+G6kXBLtM1GNkrvZvbgU2qMKa70c3IT191NJQ2wWGZl3n7SiLdhGg+veWa6qsagHKXYZgVtkzZhm21mwCttryc2cd3dZEdLLYlK2+sybT+pJNuwbSTcSlt5pNvsckfpHYTNDHhdjT+odLqNlloSdZV0+0ml2oatI+HWVa5tdlmj9A5C19FFB+iULn6iViXNF79186Tjoa0wWmopt59Unm3YYSTc0myzUY3Su2k9ePpDqi2jkpZl4MJSjpaaPtFrLSeKSrP92tumEmzDTiPhlmWbjXKUXr7JBgTGF12AwAg4EBgBBwIj4EBgBBwIjIADgRFwILD/B+BLKqb7knKzAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for epoch in range(n_epochs+1):\n",
    "    \n",
    "    with tf.GradientTape() as tape:\n",
    "        \n",
    "        # Compute persistence diagram\n",
    "        dgm = model.call()\n",
    "        \n",
    "        # Loss is sum of squared birth coordinates\n",
    "        loss = -tf.math.reduce_sum(tf.square(dgm[:,0]))\n",
    "        \n",
    "        # Compute and apply gradients\n",
    "        gradients = tape.gradient(loss, model.trainable_variables)\n",
    "        optimizer.apply_gradients(zip(gradients, model.trainable_variables))\n",
    "\n",
    "    if epoch % 10 == 0:\n",
    "        plt.figure()\n",
    "        plt.imshow(model.X.numpy())\n",
    "        plt.title(\"Image at epoch \" + str(epoch))\n",
    "        plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This upper loop looks definitely fixed now! :-)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.4"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  }
 },
 "nbformat": 4,
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}