NN_Approximation.ipynb

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   "cell_type": "markdown",
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   "source": [
    "<a href=\"https://colab.research.google.com/github/juliusberner/oberwolfach_workshop/blob/master/NN_Approximation.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "e7c__b-hItU1"
   },
   "source": [
    "This provides a framework to construct deep neural networks used to efficiently approximating various functions.\n",
    "* **modular design**: builds on simple network operations and can be easily adapted and extended\n",
    "* **versatile usage**: convenient mathematical notation and compatible with usual TensorFlow 2.x modules\n",
    "\n",
    "Theory: \n",
    "\n",
    "* [[1]](https://arxiv.org/abs/1709.05289): *Optimal approximation of piecewise smooth functions using deep ReLU neural networks* - Philipp Petersen, Felix Voigtlaender\n",
    "\n",
    "\n",
    "* [[2]](https://arxiv.org/abs/1809.07669): *DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing* - Dennis Elbrächter, Philipp Grohs, Arnulf Jentzen, Christoph Schwab\n",
    "\n",
    "\n",
    "* [[3]](https://arxiv.org/abs/1902.07896): *Error bounds for approximations with deep ReLU neural networks in $W^{s,p}$ norms* - Ingo Gühring, Gitta Kutyniok, Philipp Petersen\n",
    "\n",
    "\n",
    "* [[4]](https://arxiv.org/abs/1610.01145): *Error bounds for approximations with deep ReLU networks* - Dmitry Yarotsky\n",
    "\n",
    "\n",
    "* [[5]](https://arxiv.org/abs/1602.04485): *Benefits of depth in neural networks* - Matus Telgarsky\n",
    "\n",
    "\n",
    "* [[6]](https://arxiv.org/abs/1905.04992): *Towards a regularity theory for ReLU networks -- chain rule and global error estimates* - Julius Berner, Dennis Elbrächter, Philipp Grohs, Arnulf Jentzen \n",
    "\n",
    "\n",
    "* [[7]](https://www.sam.math.ethz.ch/sam_reports/counter/ct.php?file=/sam_reports/reports_final/reports2019/2019-07_fp.pdf): *Deep ReLU Networks and High-Order Finite Element Methods* - Joost Opschoor, Philipp Petersen, Christoph Schwab\n",
    "\n",
    "If you have any questions, please feel free to contact me:  julius.berner@univie.ac.at"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "YAxZ_FK9_cIw"
   },
   "source": [
    "## Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
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    "id": "DyP9AzjhjsLE",
    "outputId": "7e1666d6-9486-4db5-d6ec-fb259fc6b801"
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   "outputs": [],
   "source": [
    "import os\n",
    "from pathlib import Path\n",
    "\n",
    "from ipywidgets import interact\n",
    "from IPython.display import display_svg, Image, SVG\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import tensorflow as tf\n",
    "\n",
    "# using files from github repository also from colab\n",
    "module_path = Path(\"modules\")\n",
    "figures_path = Path(\"figures\")\n",
    "if not all(p.exists() and p.is_dir() for p in [module_path, figures_path]):\n",
    "    repo_path = Path('oberwolfach_workshop')\n",
    "    if not (repo_path.exists() and repo_path.is_dir()):\n",
    "        os.system(f\"git clone https://github.com/juliusberner/{repo_path.name}.git -q\")\n",
    "    os.chdir(repo_path)\n",
    "\n",
    "from modules.utils import plot_1d\n",
    "from modules import nn_constructions as nn\n",
    "from modules.nn_constructions import P, R, D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "_bt1MzTJClC7"
   },
   "source": [
    "## Neural Network Parametrizations, Realizations, and Derivatives"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "AiUYeqLe_0ae"
   },
   "source": [
    "### Parametrizations\n",
    "\n",
    "* Tutorial: [Python classes](https://docs.python.org/3/tutorial/classes.html)\n",
    "\n",
    "We follow a mathematical definition due to [P. Petersen and F. Voigtländer [1]](https://arxiv.org/abs/1709.05289):\n",
    "\n",
    "1. For a given architecture $N=(N_0,\\dots,N_L)$ define a neural network parametrization $\\Phi$ to be a finite sequence of matrix-vector pairs\n",
    "    \n",
    "   $$\\color{green} {\\Phi:=((W_k,b_k))_{k=1}^L} $$\n",
    "   \n",
    "   where  $W_k\\in\\mathbb{R}^{N_{k-1}\\times N_{k}}$ and $b_k\\in\\mathbb{R}^{N_k}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "8OEkz9UEqdMY",
    "outputId": "d2b7bb12-eaed-40d1-c195-7ee42e9d3dd4"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "my parametrization: [(array([[1., 1.]]), array([ 0., -1.])), (array([[ 1],\n",
       "       [-1]]), array([0.]))]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# shallow neural net\n",
    "a1 = np.array([[1.0, 1.0]])\n",
    "b1 = np.array([0.0, -1])\n",
    "a2 = np.array([[1], [-1]])\n",
    "b2 = np.array([0.0])\n",
    "psi = P([(a1, b1), (a2, b2)], name=\"my parametrization\")\n",
    "\n",
    "psi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 35
    },
    "id": "U6M4SAQDqdMh",
    "outputId": "87c878b8-7c3a-459b-cebf-74e16a6858ad"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Depth: 2, Width: 2, Magnitude: 1.0, #Parameters: 7, Connectivity: 5'"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi.describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "DdjIAMvqqdMl",
    "outputId": "29748805-5de9-4116-defc-61eed06f4eba"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['arch',\n",
       " 'biases',\n",
       " 'connectivity',\n",
       " 'depth',\n",
       " 'describe',\n",
       " 'eps',\n",
       " 'in_dim',\n",
       " 'magnitude',\n",
       " 'name',\n",
       " 'num_parameters',\n",
       " 'out_dim',\n",
       " 'parameters',\n",
       " 'weights',\n",
       " 'width']"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[a for a in dir(psi) if not a.startswith(\"_\")]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "PhVLwPDWqdMq",
    "outputId": "aab53f37-effc-4008-f34c-b9fecafa777d"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 2, 1]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi.arch"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "YbAYmxubqdMu",
    "outputId": "2d6a1d30-fe8d-4693-91de-60db66d41dce"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "hat: [(array([[1., 1., 1.]]), array([ 0. , -0.5, -1. ])), (array([[ 2.],\n",
       "       [-4.],\n",
       "       [ 2.]]), array([0.]))]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P(\"hat\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "rGMow099_T3J"
   },
   "source": [
    "### Realization\n",
    "* Tutorial: [Custom tf layers and models](https://www.tensorflow.org/guide/keras/custom_layers_and_models)\n",
    "\n",
    "For a given activation function $\\rho\\colon\\mathbb{R}\\to\\mathbb{R}$ define its realization $\\mathcal{R}_\\rho\\Phi\\colon \\mathbb{R}^{1\\times N_0}\\to\\mathbb{R}^{1\\times N_L}$ by \n",
    "\n",
    "  $$\\color{green}{\\mathcal{R}_\\rho\\Phi := \\mathcal{A}_L\\circ \\rho \\circ  \\mathcal{A}_{L-1}          \\circ\\dots\\circ \\rho\\circ\\mathcal{A}_1}$$\n",
    "\n",
    "  with $\\color{green}{\\mathcal{A}_k(x):=xW_k + b_k}$ and $\\rho$ applied componentwise. \n",
    "  \n",
    "  Note: In order to stay compatible with TensorFlow we multiply the input from left as a row vector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "LxQ_j_vMRTG1",
    "outputId": "1b113490-9564-4a00-dc3e-b8b94d9ba3c2"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<tf.Tensor: shape=(1, 1), dtype=float32, numpy=array([[0.5]], dtype=float32)>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "realization = R(psi)\n",
    "realization([[0.5]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "u_ht-Xq9ibEk",
    "outputId": "4d034e27-a9da-412e-9c5d-8845054d78f5"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0. ],\n",
       "       [0.5],\n",
       "       [1. ]], dtype=float32)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "realization([[0.0], [0.5], [1.0]]).numpy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "NqSZdiWCsTCk",
    "outputId": "f4d0f5c2-c10e-4219-b49c-906b0deea488"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Model: \"relu-realization of my parametrization\"\n",
      "_________________________________________________________________\n",
      "Layer (type)                 Output Shape              Param #   \n",
      "=================================================================\n",
      "layer_0 (AffineLayer)        multiple                  4         \n",
      "_________________________________________________________________\n",
      "layer_1 (AffineLayer)        multiple                  3         \n",
      "=================================================================\n",
      "Total params: 7\n",
      "Trainable params: 7\n",
      "Non-trainable params: 0\n",
      "_________________________________________________________________\n"
     ]
    }
   ],
   "source": [
    "realization.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 336
    },
    "id": "UupMVT-5RYE-",
    "outputId": "9692d265-2a4c-47cc-d98d-8176db023f2d"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "realizations = [realization, R(psi, act=tf.nn.sigmoid)]\n",
    "plot_1d(\n",
    "    *realizations, titles=[r.name for r in realizations], interval=[-1, 2], sharey=False\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "RH3MPtRR9h-i"
   },
   "source": [
    "### Derivative\n",
    "* Tutorial: [Automatic differentiation in tf](https://www.tensorflow.org/tutorials/customization/autodiff3) \n",
    "\n",
    "Define the derivative (w.r.t. the input) $\\mathcal{D}_\\rho\\Phi\\colon\\mathbb{R}^{1\\times N_0}\\to\\mathbb{R}^{N_L\\times N_0}$ by \n",
    "\n",
    "  $$\\color{green}{\\mathcal{D}_\\rho\\Phi:=W^T_L\\cdot \\operatorname{diag}(D\\rho(\\mathcal{R}^{L}_\\rho\\Phi))\\cdot W^T_{L-1}\\cdot\\ldots\\cdot \\operatorname{diag}(D\\rho(\\mathcal{R}_\\rho^{1}\\Phi))\\cdot W_1^T}$$ \n",
    "  \n",
    "  with $\\color{green}{\\mathcal{R}^K_\\rho\\Phi:=\\mathcal{R}_\\rho((W_k,b_k))_{k=1}^{K}}$ and $D\\rho$ applied componentwise.\n",
    "\n",
    "  Note: If $\\rho$ is only a.e. differentiable (i.e. locally Lipschitz continuous) we use the extended derivative \n",
    "\n",
    "  $$\\bar{D}\\rho(x) := \\begin{cases} D\\rho(x), & \\rho \\text{ is differentiable at } x \\\\ 0 &\\text{else.}  \\end{cases} $$\n",
    "\n",
    "  Then it holds that the classical derivative of the realization function  coincides with the  neural network derivative almost everywhere, i.e. \n",
    "\n",
    "  $$\\color{green}{\\mathcal{D}_\\rho\\Phi=D\\mathcal{R}_\\rho\\Phi \\quad a.e.}$$\n",
    "\n",
    "  see [[2]](https://arxiv.org/abs/1905.04992). Observe that all realizations with locally Lipschitz continuous activation function are itself locally Lipschitz continuous and a.e. classically differentiable (due to Rademacher's theorem) with its weak derivative a.e. coinciding with its classical derivative."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "p9X1MQDNqdNW",
    "outputId": "9c57a1c9-ad8c-4b03-b6f5-10e1c650988a"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.],\n",
       "       [0.]], dtype=float32)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "derivative = D(psi)\n",
    "derivative([[0.5], [0.0]]).numpy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 336
    },
    "id": "kxhbmRXML5Uk",
    "outputId": "0775d8d1-60a6-4839-b4e4-35f9d20ffbc9"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_1d(derivative, titles=[derivative.name], interval=[-1, 2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "gzyYJRHeyLGC"
   },
   "source": [
    "## Operations on Parametrizations\n",
    "\n",
    "We can define various operations on neural networks parametrizations achieving \n",
    "* Concatenation (Composition)\n",
    "* Parallelization\n",
    "* Affine Combinations\n",
    "\n",
    "of the corresponding realization functions, see [[1]](https://arxiv.org/abs/1709.05289), [[2]](https://arxiv.org/abs/1809.07669), [[3]](https://arxiv.org/abs/1902.07896).\n",
    "\n",
    "For instance, we define the concatenation with $\\Psi\\!=\\!((\\tilde{A}_k,\\tilde{b}_k))_{k=1}^{\\tilde{L}}$ by\n",
    "\n",
    "\n",
    " $$\\Psi\\circ\\Phi:=\\big(((A_k,b_k))_{k=1}^{L-1},(A_{L}\\tilde{A}_1,b_{L}^T\\tilde{A}_1 +\\tilde{b}_1),((\\tilde{A}_k,\\tilde{b}_k))_{k=2}^{\\tilde{L}}\\big)$$\n",
    "\n",
    "\n",
    "and it holds that $\\mathcal{R}_\\rho(\\Psi\\circ\\Phi) = \\mathcal{R}_\\rho(\\Psi) \\circ\\mathcal{R}_\\rho (\\Phi)$.\n",
    "\n",
    "\n",
    "\n",
    "In the following we focus on the ReLU activation function $\\color{green}{\\rho(x)=\\operatorname{ReLU}(x):=x_+}$ (and abreviate $\\mathcal{D}:=\\mathcal{D}_{\\operatorname{ReLU}}$, $\\mathcal{R}:=\\mathcal{R}_{\\operatorname{ReLU}}$).\n",
    "\n",
    "Note that such constructions could be achieved for any piecewise linear activation function with $1\\le M <\\infty$ breakpoints [[4]](https://arxiv.org/abs/1610.01145) and in an approximative fashion for any activation function that is differentiable and not constant on some open set (as this allows the uniform approximation of the identity function on a compact set)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "XEKy7IXDTDXQ"
   },
   "source": [
    "### NN operations and constructions\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 372
    },
    "id": "HyHPVbGNgtKW",
    "outputId": "299c6798-2719-4150-9ba7-85abf71b085f"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "id_1: Depth: 2, Width: 2, Magnitude: 1.0, #Parameters: 7, Connectivity: 4\n",
      "hat: Depth: 2, Width: 3, Magnitude: 2.0, #Parameters: 10, Connectivity: 8\n"
     ]
    }
   ],
   "source": [
    "pa_list = P(\"id_1\"), P(\"hat\")\n",
    "realizations = [R(pa) for pa in pa_list]\n",
    "plot_1d(*realizations, titles=[r.name for r in realizations], interval=[-1, 2])\n",
    "print(*[f\"{pa.name}: {pa.describe()}\" for pa in pa_list], sep=\"\\n\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 372
    },
    "id": "ofF1vosyhbzO",
    "outputId": "4a770fa0-10f5-4ba0-968b-0c70b56f90bf"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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+ChePH+h1JBERATJitJ8lwC+Ax06zzT+ccx+L0fFEJMqz60u599nQZ53vXT2Zj18w1ONEIiISEZOeLefca0B5LPYlIh3z962H+eJTG3AOvnbZBG6aOdLrSCIiEiWRY7bebWYbzOwFMzsvgccVSVmrdh3jzsfX0RJ03P6Bsdxx0VivI4mIyCkSVWytA0Y55y4Afg4809pGZnabmZWYWUlZWVmCookkp437K1iwtITGQJCbZo7ka5eN9zpSt6c2TERak5BiyzlX5ZyrCd9eCWSa2YBWtnvEOVfonCssKChIRDSRpLTjSDVzFhdT0xjgY+cP4T+vmoSZeR2r21MbJiKtSUixZWaDLfyXwMyKwsc9lohji6Sa/cfrmLWwmON1zVw0voD7r5tCepoKLRERv4rJ1Yhm9gRwETDAzPYD3wIyAZxzDwOfAu4wswBQD9zgnHOxOLZId1JW3cishas5VNVA0eh+PHTzdLIyNF2eiIifxaTYcs7deIbnf0FoaggR6aTK+mZuWVzM7mN1nDe0NwvnFtIjK93rWCIicgb6SCySBOqaAty6ZA1bD1YxZkBPlt5aRO+cTK9jiYhIO6jYEvG5pkCQ25evY+2e4wztk8OyBTMZ0Cvb61giItJOKrZEfKwl6LhrxXpee7OM/j2zWLZgJsPye3gdS0REOkDFlohPOee45+lNPL/pIHnZGSy9tYixBb28jiUiIh2kYkvEh5xz3PfCNlaU7CM7I41Fc2cwaVgfr2OJiEgnqNgS8aEHX9nJI6/tIiPNeHjWdIrO6ud1JBER6SQVWyI+s2zVHn704nbM4IHrp3DxhIFeRxIRkS5QsSXiI8+uL+XeZzcD8L2rJ/PxC4Z6nEhERLpKxZaIT7y07TBfemoDzsHXLpvATTNHeh1JRERiQMWWiA+s2nWMO5avIxB03P6Bsdxx0VivI4mISIyo2BLx2Kb9lSxYWkJjIMhNM0fytcvGex1JRERiSMWWiId2HKlhzqPF1DQG+Nj5Q/jPqyZhZl7HEhGRGFKxJeKR/cfrmL1oNeW1TVw0voD7r5tCepoKLRGRVKNiS8QDZdWNzFq4moOVDcwY3ZeHbp5OVoZ+HUVEUpFad5EEq6xv5pbFxew+VsfEIb1ZOGcGPbLSvY4lIiJxomJLJIHqmgLcumQNWw9WMWZATx6bX0SfHplexxIRkThSsSWSIE2BIHcsX8faPccZ2ieHZQtmMqBXttexREQkzlRsiSRAS9Bx14r1vPpmGf17ZrFswUyG5ffwOpaIiCSAii2ROHPO8Y0/bOL5TQfJy85g6a1FjC3o5XUsERFJEBVbInHknOMHL2zjyTX7yM5IY9HcGUwa1sfrWCIikkAqtkTi6MFXdvKr13aRkWY8PGs6RWf18zqSiIgkmIotkThZtmoPP3pxO2Zw//VTuHjCQK8jiYiIB1RsicTBs+tLuffZzQB89+pJXHnBUI8TiYiIV1RsicTYS9sO86WnNuAcfPWy8dw8c5TXkURExEMqtkRiaPWuY9yxfB2BoOMzHxjDnReN8zqSiIh4TMWWSIxsLq1kwdISGgNBbiwayd2XTfA6koiI+ICKLZEY2HGkhlsWF1PdGOBj5w/hu1dPwsy8jiUiIj6gYkuki/Yfr2P2otWU1zZx0fgC7r9uCulpKrRERCRExZZIF5RVNzJ7UTEHKxuYMbovD908nawM/VqJiMhJ+qsg0kmV9c3csriYt47WMnFIbxbOmUGPrHSvY4mIiM+o2BLphPqmFuYvWcPWg1WcNaAnS28tok+PTK9jiYiID6nYEumgpkCQ25evpWTPcYb0yWH5gpkU5GV7HUtERHwqJsWWmS02syNmtrmN583MfmZmO8xso5lNi8VxRRKtJei4a8V6Xn2zjH49s1g2fybD8nt4HUtERHwsVj1bS4DLTvP85cDZ4a/bgIdidFyRhHHO8c1nNvH8poPkZWfw2K1FjBvYy+tYIiLiczEptpxzrwHlp9nkKuAxF7IKyDezIbE4tkii/ODP23iieB/ZGWksnFPIpGF9vI4kIjHQ3BL0OkKrAi1BgkHndQyJgUSN2RoG7Iu6vz/8mEhSePCVHfzq1V1kpBkPz5rOzDH9vY4kIjHw9Lr9nP2NF3j1zTKvo7xNWXUjM773N77yu41eR5EY8NUAeTO7zcxKzKykrMxf//Gl+1q+ag8//PN2zOD+66dw8YSBXkcSn1IblnyeWX8AgOfC//rFC5sPcryumd+v209Dc4vXcaSLElVslQIjou4PDz/2Ns65R5xzhc65woKCggRFE2nbs+tL+fdnQ9d9fPfqSVx5wVCPE4mfqQ1LLi1Bx7o9xwEo2XO6kTCJt/Vg9Ynbr++t8C6IxESiiq3ngFvCVyW+C6h0zh1M0LFFOuWlbYf50lMbcA6+etl4bp45yutIIhJD2w5VUdMYAGDPsTqOVDd4nOiktVHF3z93HfMwicRCRix2YmZPABcBA8xsP/AtIBPAOfcwsBK4AtgB1AHzYnFckXhZvesYdyxfRyDo+MwHxnDnReO8jiQiMVay+/jb7q/dfZzLJ3t/7VZFXRNvHq45cX+Viq2kF5Niyzl34xmed8BnY3EskXjbXFrJgqUlNAaC3Fg0grsvm+B1JBGJgzW7Q71Hw/J7UFpRzxqfFFtrw6c2zxnUizcP17BhXwVNgaDWXU1i+smJRNlxpIZbFhdT3Rjgo+cP4btXT8bMvI4lIjHmnDtRbC248CzAP+O21oR73C45dxBnD+xFYyDI5gOVHqeSrlCxJRJWWlHP7EWrKa9t4gPnFPDAdVNIT1OhJZKK9h+v53BVI/m5mXxq+nDS04wtB6qoawp4He3EeK0Zo/tROLofACW7/VEISueo2BIBjtY0Mnvhag5WNlA4qi8Pz5quLnuRFBbpxSoc1Ze8nEwmDulNS9Cx3uMr/xqaW9iwrxIzmDayL4Wj+gIne7skOemviXR7lfXN3LKomF1Ha5k4pDeL5s6gR1a617FEJI4ixUuk56hwtD+Kms2llTS1BDlnYB59cjOZEc63ds9xQsOfJRmp2JJurb6phQVL1/DGwSrOGtCTpbcW0adHptexRCTOIqflZoSLrMJR4dN1Ho/bOlkEhnKN6NeDgXnZlNc2setorZfRpAtUbEm31RQIcvvytazZfZwhfXJYvmAmBXnZXscSkTiLTK2QlZF2Yo3TSHGzbs9xAh6ulRg9XgvAzE7c1rit5KViS7qllqDjrqfW8+qbZfTrmcWy+TMZlt/D61gikgCRqRWmDM8nOyM0ZGBQ7xxG9sultqmFbYeqT/fyuAkGHSV73t6zBTBd47aSnoot6Xacc3zzmU08v/EgvbIzeOzWIsYN7OV1LBFJkFNP1UVEBqN71YO0s6yGirpmBvfOeduHv+hxW5KcVGxJt/ODP2/jieJ9ZGeksWhO4YnTCCLSPZwcr9XvbY+fmGbBo6Imulcren6/c4fkkZuVzltHaymrbvQkm3SNii3pVh58ZQe/enUXGWnGQ7OmMXNMf68jiUgCNTS3sHF/aILQaSPf3rM148QVieWeXPm3po0iMCM9jakj84G3r5koyUPFlnQbj6/eww//vB0z+Ml1F/DBCYO8jiQiCbYpPLXC+EGhqRWijS3oRX5uJoerGtl/vD7h2UraOL0JJ6+W1Lit5KRiS7qF5zYc4JvPbAbgP6+axFVThnmcSES8EOk9aq2gSUszpod7uxI9BcSRqgb2ltfRKzuDCYN7v+P5GR6f4pSuUbElKe/lbUf44or1OAdfuXQ8s941yutIIuKRSO/RqafqIk4uj5PYoiZSRE0dmd/qMmFTwo9vKa30xZJC0jEqtiSlFb9Vzu3L1xIIOj7z/jHcedFYryOJiEeCQXdicHxrPVtwctxWoouttsZrRfTKzuDcIXkEgo71+yoSmExiQcWWpKzNpZXMX7KGxkCQG4tGcPflE952hY+IdC//OlJDVUOAIX1y2pxXb9KwPmSlp7H9cDWVdc0Jy3ZivNao1ovA0HPe9LpJ16nYkpS0s6yGOYuLqW4M8NHzh/Ddqyer0BLp5k6O1+rXZnuQk5nO+cND08Gs3ZuYcVu1jQHeOFhFepoxJXzVYWs0bit5qdiSlFNaUc/shas5VtvEB84p4IHrprQ6BkJEupfIpKAz2jiFGJHocVvr91XQEnRMGtqb3KyM0+Q6uaRQS1CLUicTFVuSUo7WNDJ74WoOVDZQOKovD82aRlaG/puLSFTP1qjWx0VFJHrcVnSP2+kM6p3DiH49qGkMsO1QVSKiSYzor5CkjMr6Zm5ZVMyuo7WcO6Q3i+bOOO2nRBHpPg5W1rP/eD152RmMH5x32m0jaxGu319BY6Al7tnaM14rYsYoLd2TjFRsSUqob2phwdI1vHGwirMG9OSxW4vo0yPzzC8UkW4hUtBMG9X3jMMK8nOzOHtgL5oCQTaXxrcHKdAS5PW9oWzTz3B6E072fmly0+SiYkuSXlMgyB2Pr2XN7uMM6ZPDsvlFFORlex1LRHzk5HqIZy5oIHrcVnwHyW87VE1tUwuj++cyMC+nHbnCSwq95c2SQtI5KrYkqbUEHV98aj2vbC+jX88sls2fyfC+uV7HEhGfWXNiKZzTj4uKiJzSi3cPUmS81vQzjCOLGFfQiz49MjlU1UBpReKXFJLOUbElScs5xzef2cyfNh6kV3YGS+cVMW5gL69jiYjPVDU0s+1QFZnpxgXD89v1msg0C2v3xLcH6eSM9u3rcUtLsxOFoMZtJQ8VW5K0/uvP23mieC/ZGWksmlPI5PDcOCIi0V7fW0HQhSYs7ZGV3q7XjOjXg4F52Ryva2ZnWW1ccjnnTqzB2N4et+ht18T5FKfEjootSUoPvbKTh1/dSUaa8dCsacwc09/rSCLiUyVnWAqnNWZ2chLROBU1+4/Xc7iqkb65mYwt6Nnu1xV6tKSQdJ6KLUk6j6/ew3/9eRtm8JPrLuCDEwZ5HUlEfOzk/FrtO1UXMT3O47aix2t1ZIWLydFLCtUnbkkh6TwVW5JUnttwgG8+sxmA71w1iaumDPM4kYj4WVMgeGLh5ukdLLaix23FQ0k7Z7Q/VWRJIedg3V71biUDFVuSNF7edoQvrliPc/CVS8cz+12jvI4kIj635UAlDc1Bxhb0pH+vjk0Jc+6QPHKz0tl9rI4j1Q0xz1bSzpnjWzP9xKlEjdtKBiq2JCkUv1XO7cvXEgg6bnv/GO68aKzXkUQkCZy82q/jBU1GehrTRoav/IvxqcSKuibePFxDVkYak4b17vDrIzPJa3LT5KBiS3xvc2kl85esoTEQ5IYZI/j65RM6NL5BRLqv9q472JZ4jduKTNswZXg+2Rntu0KytVwb9lXQFAjGNJvEnoot8bWdZTXMWVxMdWOAj04ewveumaxCS0TaJTS1QufGRUXEa9xWJFdhJ3P17RlaUqgxEGTzgcpYRpM4ULElvlVaUc/shas5VtvE+88p4IHrp5xxTTMRkYhdR2spr22iIC+bkf06t7LElJH5pKcZmw9UUdcUiFm2zkxHcapCjdtKGjEptszsMjPbbmY7zOzuVp6fa2ZlZrY+/LUgFseV1HW0ppHZC1dzoLKBwlF9eXjWNLIy9NlARNovej3EzvaI98rO4NwhebQEHev3VsQkV0NzCxv2hXqjImPCOqNQ47aSRpf/eplZOvBL4HJgInCjmU1sZdMVzrkp4a+FXT2upK6qhmbmLC5m19Fazh3Sm0VzZ5CbleF1LBFJMifWQ2znuoNtiXVRs7m0kqaWIOMH5dEnN7PT+zl5ivO4FqX2uVh0FRQBO5xzu5xzTcCTwFUx2K90Q/VNLcxfsoYtB6o4a0BPHru1iD49Ot8YiUj3FYtTddGvL4nRuK2ujteKiCwpVF7bxK6j8VlSSGIjFsXWMGBf1P394cdO9Ukz22hmvzOzETE4rqSYpkCQOx5fy5rdxxncO4dl84soyOvYvDgiIgBHqhvYfayO3Kx0zh2S16V9RYqidXuOE2jp+pV/sSoCzUzjtpJEogbB/BEY7Zw7H/grsLS1jczsNjMrMbOSsrKyBEUTP2gJOr741Hpe2V5Gv55ZLF9QxPC+nRvQKuIVtWH+EZkXa9rIvmSkd+1P3aDeOYzo14Papha2Haru0r6CwZNXSHZ0RvvWaNxWcohFsVUKRPdUDQ8/doJz7phzrjF8dyEwvbUdOececc4VOucKCwoKYhBNkoFzjn9/djN/2niQXtkZLJ1XxLiBXfskKuIFtWH+cWK8VhdP1UVEJhGNzI/VWbuO1lBR18zg3jkM79uj67lGxyaXxFcsiq01wNlmdpaZZQE3AM9Fb2BmQ6LuXglsjcFxJUX88MXt/Gb1XrIz0lg4p5DJw/t4HUlEklxkfFVXB8dHRCZFXdPF03XRRWAs5gyMLCn01tFayqobz/wC8USXiy3nXAD4HPAioSLqKefcFjP7jpldGd7s82a2xcw2AJ8H5nb1uJIaHn51Jw+9spOMNOPBm6fxrjH9vY4kIkmutjHAlgNVpKcZU0bmx2SfkR6yNbvLu3Tl34kZ7WNwChFCSwpNDX+P8VowW7ouJmO2nHMrnXPnOOfGOue+F37sXufcc+HbX3fOneecu8A5d7FzblssjivJ7Ter9/KDF7ZhBj+57gI+dO4gryOJSApYv6+ClqBj4pDe9MqOzbQx4wp60adHJoerGtl/vL7T+yk50bMVmx43ONl7V6JxW76lWSLFE3/ccIBvPLMJgO9cNYmrprR2AauISMedXA8xNr1HAGlpdqI3qrPjo45UNbC3vI5e2RlMGBy7camRcVtrNG7Lt1RsScK9vP0Id61Yj3PwlUvHM/tdo7yOJCIpJNLD09WpFU7V1XFbkasQp47M7/IVktGmjMwnzWBLaWVMlxSS2FGxJQlV/FY5dyxfSyDouO39Y7jzorFeRxKRFBJoCbJub2Tm+Nj1bEH0WoSd60E6OV4rtkVgr+wMJg7tTSDoWL+vIqb7lthQsSUJs7m0kvlL1tDQHOT6whF8/fIJMbkaR0QkYuvBauqaWhjVP5eBvXNiuu/Jw/qQlZ7G9sPVVNY1d/j1J3vcYlsEwskCbq3GbfmSii1JiJ1lNcxZXEx1Y4ArJg/m+5+YrEJLRGIuXr1HADmZ6Zwfnpom0nvWXrWNAd44GNsrJKOduFpS47Z8ScWWxN2BinpmL1zNsdomLjx7AA9cP4X0NBVaIhJ7kfm14tF7BDA9agqIjohcIXne0N7kZsXmCslokeJy3Z7jtAS1KLXfqNiSuDpa08isRas5UNnA9FF9+dXs6WRnpHsdS0RSkHMuatLQ2PdswcmZ5Ds6biuePW4Ag/uElhSqaQywvYtLCknsqdiSuKlqaGbO4mJ2ldUyYXAei+fMiMsnOhERgL3ldZRVN9I3N5OxBT3jcozIeoYb9lfQGGhp9+si00XEq8cNogpBTW7qOyq2JC7qm1pYsKSELQeqGN0/l2XzZ9InN9PrWCKSwqInDI3XmNC+PbM4e2AvGgNBNpdWtes1gZYg6yKLT8ex2Dp5ilPjtvxGxZbEXFMgyJ2Pr6V4dzmDe+ewfMFMCvKyvY4lIiku3uO1Ik5OAdG+HqRth6qpjVwhmRfbKySjReYVa28uSRwVWxJTLUHHl367gZe3l9GvZxbLFxQxvG+u17FEpBuI93itiMi4q/b2IMV7vFZEZEmhg5UNlFZ0fkkhiT0VWxIzzjn+/dnN/HHDAXplZ7B0XhHjBsZuSQoRkbaU1zax40gN2RlpTBraJ67HivQgrd3TvkWpSxIwXgvevqSQerf8RcWWxMwPX9zOb1bvJTsjjYVzCpk8PL4NnohIRGQA+pQR+WRlxPdP24h+PRiYl83xumZ2ltWedlvn3InCJ949btD5qSkkvlRsSUw8/OpOHnplJ+lpxoM3T+NdY/p7HUlEupFIQRPr9RBbY2btHre1/3g9h6vie4VktJPjtjRI3k9UbEmXPVG8lx+8sA0z+Mm1F/Chcwd5HUlEupkT46LifKouovDENAunL2oig/anj4rfFZLR3rakUH3HlxSS+FCxJV3yp40HuOcPmwD4zpXncfXUYR4nEpHupqG5hU2llZjBtBgvPt2W9l75tyaO6yG2JiczncnD++Bcx5cUkvhRsSWd9sr2I9y1Yj3OwZc/cg6z3z3a60gi0g1t2FdBc4tjwuDe9M5JzHx+5w7JIzcrnd3H6jhS3dDmdiUJ7nGLPpYGyfuHii3plDW7y7l9+VqaWxyfvvAsPnvxOK8jiUg3lair/aJlpKcxNbyg9No2xkdV1DXx5uEasjLSmDQscRcMdXZJIYkfFVvSYVsOVHLrkjU0NAe5vnAE91xxbkLGIoiItGZNAq/2i3amcVuR03hThucndE3YyJJC6/dV0BQIJuy40jYVW9Ihu8pquGVRMdUNAa6YPJjvf2KyCi0R8UxL0CVk3cHWnGnc1slJVhObq2/PLMZFlhQ6UJnQY0vrVGxJux2oqGfWwtUcq23iwrMH8MD1U0hPU6ElIt5583A11Q0BhuX3YEifHgk99pSR+aQZbD5QRV1T4B3PezFeK2KGxm35iootaZejNY3MWrSaA5UNTB/Vl1/Nnp7QbnERkdacnF8r8QVNr+wMJg7tTUvQsX5vxdueawy0sGF/qFdp+sjEnt6EqFOcGrflCyq25IyqGpqZs7iYXWW1TBicx+I5M8jNyvA6lohIwtZDbEtb47Y2l1bSFAgyflAefXITc4VktBOnOPccb9eSQhJfKrbktBqaW1iwpIQtB6oY3T+Xx+YXedJwiIi0JpEzx7emsI3lcSJF4HQPetwgtKRQQV425bVN7Dp6+iWFJP5UbEmbmluC3Pn4Oop3lzO4dw7L5s9kYF6O17FERAAorajnQGUDvXMyOHtgL08yRHq2Xt9bQUvwZA+Sl6c3IbSkkMZt+YeKLWlVS9Dxpac28NK2I/TNzWT5giJG9Mv1OpaIyAnRCzyneXSxzuA+OYzo14OaxgDbDlUBEIy6QjJSjHlB47b8Q8WWvINzjnuf3cxzGw7QKzuDpbcWMW5gntexRETeJnLqbnqCluhpy6mTiO46WsPxumYG985heN/EXiEZ7cRM8mdYv1HiT8WWvMOPXtzO46v3kpWRxq9vKeT84fleRxIReYeSE+sOetd7BCfHZUWKv+jxWl7OQzhxSG9ys9J562gtZdWNnuUQFVtyil+9upMHX9lJeprx4E3TePfY/l5HEhF5h8q6ZrYfriYrPY3zhyduKZzWnJzcNHTlX6TomuFxj9vblhRS75anVGzJCU8U7+W+F7ZhBj+59gIumTjI60giIq1at/c4zsHk4X3IyfR2zr9xBb3o0yOTQ1UNlFbUnxyv5XGPG0SP29IgeS+p2BIA/rTxAPf8YRMA37nyPK6eOszjRCIibVvj4ezsp0pLsxPjxlZuOsieY3X0zEpnwmDvx7qemJpCPVueUrElvLL9CHetWI9z8OWPnMPsd4/2OpKIyGmdGK/l4dV+0SJFza9e3QXAtFF9yUj3/k/s1JF9STPYUlpJfVOL13G6rZj8TzCzy8xsu5ntMLO7W3k+28xWhJ9fbWajY3Fc6bqS3eXcvnwtzS2OT194Fp+9eJzXkURETqsx0ML6/RWA91ciRkTGbR2rbQK8nfIhWmRJoUDQsX5fhddxuq0uF1tmlg78ErgcmAjcaGYTT9lsPnDcOTcOeAD4r64eV7puy4FK5i1ZQ0NzkOsKh3PPFed6euWMiEh7RJbCOXtgL/r2zPI6DgCTh/UhK6ony6vJTFujcVvei0XPVhGwwzm3yznXBDwJXHXKNlcBS8O3fwd8yPRX3VO7ymqYs7iY6oYAl08azH2fOF+FlogkhVW7/DNeKyInM51zh5wcozUlfBWgH0Tep1VvHfM4SfcVi9WEhwH7ou7vB2a2tY1zLmBmlUB/4GgMji8ddKCintmLijla08SFZw/gv2+YQrpHsy+LdHd3/34jf3njsNcxkkpNQwCAi8cP9DjJ210zdRgb9lfy4YmDyM2KxZ/X2Hjv2AFkpBn/u+MY0/7zr17HSSr5PTJ56csXdXk//vnfAJjZbcBtACNHjvQ4TWo6VtPIrEWrKa2oZ9rIfH41ezrZGd5eNi2SKjrThtU0BigPj/OR9hvdP5cPjC/wOsbbzH3vWVwzbTi9c3z1p5W+PbO4Zuowfrt2v/6vdZBz7swbtYN1dUdm9m7g2865S8P3vw7gnLsvapsXw9v808wygENAgTvNwQsLC11JSUmXssnbVTU0c9OvV7G5tIoJg/NYcdu76ZOb6XUskRPMbK1zrtDrHLHQ3jasuqGZpkAwAYlSS58emb642i9ZOOc4Xtccs+KhuzAz+rVzXODp2q9YlN9rgLPN7CygFLgBuOmUbZ4D5gD/BD4FvHS6Qktir6G5hQVLS9hcWsXo/rk8Nr9IhZaID+Tl6PdQ4q8jRYPEXpeLrfAYrM8BLwLpwGLn3BYz+w5Q4px7DlgELDOzHUA5oYJMEqS5Jcidj6+j+K1yBvfOYdn8mQzMy/E6loiISLcQkxPLzrmVwMpTHrs36nYDcG0sjiUd0xJ0fOmpDby07Qh9czNZvqCIEf1yvY4lIiLSbeiEdwpzzvGt5zbz3IYD9MxKZ+mtRYwb6P3yESIiIt2Jiq0U9uO/bGf5qr1kZaSxcM4Mzh+e73UkERGRbkfFVop65LWd/PLlnaSnGQ/eNI13j+3vdSQREZFuScVWCnqyeC/fX7kNgB9fez6XTBzkcSIREZHuS8VWinl+40G+/odNAHznqvO4ZupwjxOJiIh0byq2Usgr24/wbytexzn40ofP4ZZ3j/Y6koiISLenYitFlOwu5/bla2lucSx431l87oPjvI4kIiIiqNhKCVsOVDJvyRoamoNcVzicb3z0XMy0sLSIiIgfqNhKcrvKapizuJjqhgCXTxrMfZ84X4WWiIiIj6jYSmIHK+uZvaiYozVNXHj2AP77himkp6nQEhER8RMVW0nqWE0jsxauprSinqkj83l41nSyM9K9jiUiIiKnULGVhKobmpn76Bp2ltUyYXAeS+YW0TM7JstcioiISIyp2EoyDc0tzF9awqbSSkb1z+Wx+UX0yc30OpaIiIi0QcVWEmluCXLn4+sofqucQb2zWT5/JgPzcryOJSIiIqehYitJBIOOL/92Ay9tO0Lf3EyWz5/JiH65XscSERGRM1CxlQScc9z73GaeXX+AnlnpLJlXxNmD8ryOJSIiIu2gYisJ/Pgv21m+ai9ZGWksnDODC0bkex1JRERE2knFls898tpOfvnyTtLTjAdvmsa7x/b3OpKIiIh0gIotH3uyeC/fX7kNgB9fez6XTBzkcSIRERHpKBVbPrVy00Hu+cMmAP7jyvO4ZupwjxOJiIhIZ6jY8qFX3yzjC0++TtDBFz98DnPeM9rrSCIiItJJKrZ8Zu2ecm5ftpbmFsf8953F//vgOK8jiYiISBeo2PKRNw5UMffRNdQ3t3Dt9OF886PnYqaFpUVERJKZii2feOtoLbcsXk11Q4DLzhvMfZ+YrEJLREQkBajY8oGDlfXMWriaozVNXHj2AH564xQy0vWjERERSQX6i+6xYzWNzFq4mtKKeqaOzOfhWdPJzkj3OpaIiIjEiIotD1U3NDP30TXsLKtlwuA8lswtomd2htexREREJIZUbHmkobmFBUtL2FRayaj+uTw2v4g+uZlexxIREZEYU7HlgeaWIJ99fB2r3ypnUO9sls+fycC8HK9jiYiISByo2EqwYNDx5d9u4O/bjpCfm8my+TMZ0S/X61giIiISJyq2Esg5x7ee28Kz6w/QMyudpfOKOGdQntexREREJI5UbCXQT/7yJstW7SErI41fzynkghH5XkcSERGROOtSsWVm/czsr2b2r/C/fdvYrsXM1oe/nuvKMZPVr1/bxS9e3kF6mvHLm6bxnrEDvI4kIiIiCdDVnq27gb87584G/h6+35p659yU8NeVXTxm0lmxZi/fW7kVgB9fez4fnjjI40QiIiKSKF0ttq4CloZvLwWu7uL+Us7KTQf5+tObAPiPK8/jmqnDPU4kIiIiidTVYmuQc+5g+PYhoK0umxwzKzGzVWZ2dRePmTRefbOMLzz5OkEHX/zwOcx5z2ivI4mIiEiCnXG6cjP7GzC4lae+EX3HOefMzLWxm1HOuVIzGwO8ZGabnHM7WznWbcBtACNHjjxjeD9bu6ec25etpbnFMf99Z/H/PjjO60giEmep1IaJSOycsdhyzl3S1nNmdtjMhjjnDprZEOBIG/soDf+7y8xeAaYC7yi2nHOPAI8AFBYWtlW4+d7Wg1XMe3QN9c0tXDt9ON/86LmYmdexRCTOUqUNE5HY6uppxOeAOeHbc4BnT93AzPqaWXb49gDgvcAbXTyub711tJbZi4qpaghw2XmDue8Tk1VoiYiIdGNdLbZ+AHzYzP4FXBK+j5kVmtnC8DbnAiVmtgF4GfiBcy4li62DlfXMWriaozWNvG/cAH564xQy0jWVmYiISHd2xtOIp+OcOwZ8qJXHS4AF4dv/B0zuynGSQXltE7MXFVNaUc/Ukfn8avZ0sjPSvY4lIiIiHlO3SwxUNzQzZ3ExO47UMH5QHo/OnUHP7C7VsSIiIpIiVGx1UUNzCwuWlrCptJJR/XNZNr+I/Nwsr2OJiIiIT6jY6oLmliCffXwdq98qZ1DvbJbPn8nA3jlexxIREREfUbHVScGg48u/3cDftx0hPzeTZfNnMqJfrtexRERExGdUbHWCc45v/3ELz64/QM+sdJbOK+KcQXlexxIREREfUrHVCff/9U0e++cesjLS+PWcQi4Yke91JBEREfEpFVsdtPAfu/j5SztITzN+ceNU3jN2gNeRRERExMdUbHXAU2v28d3ntwLww0+ez0fOa23JSBEREZGTVGy108pNB7n76Y0AfPvjE/nk9OEeJxIREZFkoGKrHV57s4wvPPk6QQd3XXIOc997lteRREREJEmo2DqDtXvK+cyytTS3OG5971l8/kPjvI4kIiIiSUTF1mlsPVjFvEfXUN/cwqemD+ebHz0XM/M6loiIiCQRFVtteOtoLbMXFVPVEOCy8wbzg09MJi1NhZaIiIh0jIqtVhyqbGDWwtUcrWnkfeMG8NMbp5CRrrdKREREOk4VxCnKa5uYtWg1pRX1TB2Zz69mTyc7I93rWCIiIpKkVGxFqW5oZu6jxew4UsP4QXk8OncGPbMzvI4lIiIiSUzFVlhDcwuffqyEjfsrGdkvl2Xzi8jPzfI6loiIiCQ5FVtAc0uQz/1mHat2lTMwL5vHF8xkYO8cr2OJiIhICuj2xVYw6Pjq7zbyt61HyM/NZPmCmYzol+t1LBEREUkR3brYcs7x7T9u4Q+vl9IzK50l84o4Z1Ce17FEREQkhXTrYuv+v77JY//cQ1ZGGr++pZApI/K9jiQiIiIpptsWWwv/sYufv7SD9DTjFzdO5T3jBngdSURERFJQtyy2nlqzj+8+vxWAH37yfD5y3mCPE4mIiEiq6nbF1gubDnL30xsB+PbHJ/LJ6cM9TiQiIiKprFsVW//4VxlfeHI9QQd3XXIOc997lteRREREJMV1m2Jr7Z7j3PbYWppagsx772g+/6FxXkcSERGRbqBbFFtbD1Yx79Fi6ptb+NT04fz7RydiZl7HEhERkW4g5Yut3Udrmb2omKqGAJeeN4gffGIyaWkqtERERCQxUrrYOlTZwM0LV3O0ppH3juvPT2+YSkZ6Sn/LIiIi4jMpW3mU1zYxa9FqSivqmTIin0dmF5KTme51LBEREelmUrLYqm5oZu6jxew4UsP4QXksmTeDntkZXscSERGRbijliq2G5hY+/VgJG/dXMrJfLsvmF5Gfm+V1LBEREemmUqrYam4J8rnfvM6qXeUMzMvm8QUzGdg7x+tYIiIi0o11qdgys2vNbIuZBc2s8DTbXWZm281sh5nd3ZVjtiUYdHz1dxv529bD5OdmsnzBTEb0y43HoURERETaras9W5uBTwCvtbWBmaUDvwQuByYCN5rZxC4e922cc/zHH7fwh9dLyc1KZ8m8Is4ZlBfLQ4iIiIh0SpdGjTvntgJnmiC0CNjhnNsV3vZJ4Crgja4cO9oDf32Tpf/cQ1Z6Gr++pZApI/JjtWsRERGRLknEmK1hwL6o+/vDj72Dmd1mZiVmVlJWVtaunW8ureRnL+0gPc34+U1Tee+4AV1PLCLSCZ1pw0Qk9Z2xZ8vM/gYMbuWpbzjnno1lGOfcI8AjAIWFha49r5k0rA8/+tT5mBmXntdaTBGRxOhMGyYiqe+MxZZz7pIuHqMUGBF1f3j4sZi5tnDEmTcSERER8UAiTiOuAc42s7PMLAu4AXguAccVERER8VxXp364xsz2A+8GnjezF8OPDzWzlQDOuQDwOeBFYCvwlHNuS9dii4iIiCSHrl6N+AfgD608fgC4Iur+SmBlV44lIiIikoxSagZ5EREREb9RsSUiIiISRyq2REREROJIxZaIiIhIHKnYEhEREYkjFVsiIiIicaRiS0RERCSOzDl/Lt9lZmXAng68ZABwNE5xusKvucC/2fyaC/ybza+5oGPZRjnnCuIZJlE62Ialys8vkfyaC/ybTbk6Libtl2+LrY4ysxLnXKHXOU7l11zg32x+zQX+zebXXODvbH7h5/fIr9n8mgv8m025Oi5W2XQaUURERCSOVGyJiIiIxFEqFVuPeB2gDX7NBf7N5tdc4N9sfs0F/s7mF35+j/yaza+5wL/ZlKvjYpItZcZsiYiIiPhRKvVsiYiIiPhO0hZbZnatmW0xs6CZtXmlgJldZmbbzWyHmd2dgFz9zOyvZvav8L9929iuxczWh7+ei3Om074HZpZtZivCz682s9HxzNOBXHPNrCzqfVqQoFyLzeyImW1u43kzs5+Fc280s2k+yXWRmVVGvV/3JijXCDN72czeCP9OfqGVbTx5z/zKr+1X+Ji+asPUfnU4ly/br3ZmS902zDmXlF/AucB44BWgsI1t0oGdwBggC9gATIxzrh8Cd4dv3w38Vxvb1STofTrjewDcCTwcvn0DsMInueYCv/Dg/9b7gWnA5jaevwJ4ATDgXcBqn+S6CPiTB+/XEGBa+HYe8GYrP0tP3jO/fvm1/Qof1zdtmNqvTmXzZfvVzmwp24Ylbc+Wc26rc277GTYrAnY453Y555qAJ4Gr4hztKmBp+PZS4Oo4H+9M2vMeRGf+HfAhMzMf5PKEc+41oPw0m1wFPOZCVgH5ZjbEB7k84Zw76JxbF75dDWwFhp2ymSfvmV/5uP0Cf7Vhar86yK/tVzuzeSIRbVjSFlvtNAzYF3V/P+98A2NtkHPuYPj2IWBQG9vlmFmJma0ys6vjmKc978GJbZxzAaAS6B/HTO3NBfDJcJft78xsRJwztZcX/6/a691mtsHMXjCz8xJ98PApnKnA6lOe8vN75ldevWd+asPUfsWe338XU7INy+hysjgys78Bg1t56hvOuWcTnSfidLmi7zjnnJm1dbnnKOdcqZmNAV4ys03OuZ2xzprk/gg84ZxrNLPPEPr0+kGPM/nZOkL/r2rM7ArgGeDsRB3czHoBvwf+zTlXlajj+pVf2y9QG5Ygar86LmXbMF8XW865S7q4i1Ig+tPE8PBjXXK6XGZ22MyGOOcOhrsYj7Sxj9Lwv7vM7BVClXQ8Gqr2vAeRbfabWQbQBzgWhywdyuWci86wkNBYEj+Iy/+rropuHJxzK83sQTMb4JyL+5pjZpZJqJF63Dn3dCub+PI9iye/tl+QVG2Y2q/Y8+3vYiq3Yal+GnENcLaZnWVmWYQGT8b1yr/w/ueEb88B3vEJ1sz6mll2+PYA4L3AG3HK0573IDrzp4CXXHhEYBydMdcp58OvJHQe3Q+eA24JX53yLqAy6rSLZ8xscGSsipkVEfr9jvcfHcLHXARsdc7d38ZmvnzPfM6L9gv81Yap/Yo93/4upnQb1pmR+374Aq4hdM60ETgMvBh+fCiwMmq7KwhdWbCTUPd9vHP1B/4O/Av4G9Av/HghsDB8+z3AJkJXsGwC5sc50zveA+A7wJXh2znAb4EdQDEwJkE/wzPlug/YEn6fXgYmJCjXE8BBoDn8f2w+cDtwe/h5A34Zzr2JNq4m8yDX56Ler1XAexKU632AAzYC68NfV/jhPfPrl1/br/AxfdWGqf3qcC5ftl/tzJaybZhmkBcRERGJo1Q/jSgiIiLiKRVbIiIiInGkYktEREQkjlRsiYiIiMSRii0RERGROFKxJSIiIhJHKrZERERE4kjFloiIiEgc/X+AhWtyhfmdlQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 720x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "id_1 | Depth: 3, Width: 2, Magnitude: 1.0, #Parameters: 13, Connectivity: 8\n",
      "hat ○ hat | Depth: 3, Width: 3, Magnitude: 2.0, #Parameters: 22, Connectivity: 19\n"
     ]
    }
   ],
   "source": [
    "# elongation of identity to depth 3 and composition of hat function with itself\n",
    "new_pa_list = nn.elongation(pa_list[0], 3), nn.conc([pa_list[1], pa_list[1]])\n",
    "realizations = [R(pa) for pa in new_pa_list]\n",
    "plot_1d(*realizations, titles=[r.name for r in realizations], interval=[-1, 2])\n",
    "print(*[f\"{pa.name} | {pa.describe()}\" for pa in new_pa_list], sep=\"\\n\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 354
    },
    "id": "RKtQoxAukRkC",
    "outputId": "d47a73b3-a8d7-4394-ef24-852433b6b903"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Depth: 2, Width: 5, Magnitude: 2.0, #Parameters: 16, Connectivity: 12\n"
     ]
    }
   ],
   "source": [
    "pa = nn.affine_combination([1.0, 1.0], pa_list=pa_list, ind_list=[[0], [0]])\n",
    "# addition of realization functions -> outputs id(x) + hat(x)\n",
    "realization = R(pa)\n",
    "plot_1d(realization, titles=[realization.name], interval=[-1, 2])\n",
    "print(pa.describe())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "kwwVyQc3leY2",
    "outputId": "ee3d7c42-1742-42a5-b7d3-a8766f26e930"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2, 5, 1]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# addition of realization functions for separate inputs (x_1, x_2) -> outputs id(x_1) + hat(x_2)\n",
    "pa = nn.affine_combination([1.0, 1.0], pa_list=pa_list)\n",
    "pa.arch"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "AcnEkWNilkZH",
    "outputId": "a365b9ec-c52c-44a1-fcda-0997b60fa7f9"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.75]], dtype=float32)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R(pa)([[0.25, 0.25]]).numpy()  # 2-dimensional input"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "dTIU10PgqdOY",
    "outputId": "fa18395a-119e-43e0-d649-90bf59fb1556"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.  ],\n",
       "       [0.75]], dtype=float32)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R(pa)(\n",
    "    [[1.0, 0.0], [0.25, 0.25]]\n",
    ").numpy()  # or general: inputs of dimension [batch_size, 2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "6ok-pkt7iS4n",
    "outputId": "2681c841-e084-43b8-93b2-0577ace3a9cd"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 2.]], dtype=float32)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D(pa)([[0.25, 0.25]]).numpy()  # derivative of id and hat"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "yI69JK3B3uGI"
   },
   "source": [
    "## Constructive Approximation Results (in $W^{1,\\infty}$ norm)\n",
    "\n",
    "In the following we show how to approximate a richer class of functions by ReLU networks with \"almost optimal\" rates (in the spirit of Yarotsky [[4]](https://arxiv.org/abs/1610.01145))."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "XssoiGlO4Phn"
   },
   "source": [
    "### Telgarsky Saw-Tooth Construction \n",
    "\n",
    "*For any positive integer k, there exist neural networks with $\\mathcal{O}(k^3)$  layers, $\\mathcal{O}(1)$ nodes per layer, and $\\mathcal{O}(1)$ distinct parameters which can not be approximated by networks with $\\mathcal{O}(k)$ layers unless they are exponentially large --- they must possess $\\Omega(2^k)$ nodes [[7]](https://arxiv.org/abs/1602.04485).*\n",
    "\n",
    "Each application of the triangle ReLU network doubles the number of \"theeth\"  $\\Rightarrow$  $\\color{green}{2^k}$ teeth with $\\color{green}{\\mathcal{O(k)}}$ network size (layers/parameters)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 368,
     "referenced_widgets": [
      "65874f5747a940a7b59243c4c541ab60",
      "c0c54cc0f7b744adbb95f2aae003f26f",
      "cc036f78ad9b4e569b01c127aadfb875",
      "8341fa7a557a4350aa1631847271dadd",
      "4415426de90447e48453aa7e534cbd03",
      "9f63eb9b329447c3a04f074fd3ed9f16",
      "a0d3a5bfadc849dd8158f2d9c9fdb506"
     ]
    },
    "id": "WRvct0yj8E1T",
    "outputId": "c6cb870c-3d28-419e-9d33-64f6eb9745de"
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "2c4bc43621d0479aa476b72dad1291e1",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=3, description='k', max=5, min=1), Output()), _dom_classes=('widget-inte…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_sawtooth(k):\n",
    "    realization = R(nn.conc([P(\"hat\") for _ in range(k)]))\n",
    "    plot_1d(realization, titles=[realization.name], interval=[-0.5, 1.5], num=int(1e5))\n",
    "\n",
    "\n",
    "interact(plot_sawtooth, k=(1, 5));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "5C34l_K_9_3a"
   },
   "source": [
    "### Yarotsky Squaring Network \n",
    "\n",
    "*Proposition 2. The function $f (x) = x^2$ on the segment [0, 1] can be approximated with any\n",
    "error $\\varepsilon > 0$ by a ReLU network having the depth and the number of weights and computation\n",
    "units $\\mathcal{O}(ln(1/\\varepsilon))$ [[6]](https://arxiv.org/abs/1610.01145).*\n",
    "\n",
    "$\\big($ lin. interp. of $f$ on intervals of size $2^{-k}$ $\\big)$ - $\\big($ lin. interp. on intervals of size $2^{-(k+1)}$  $\\big) = \\big($ scaled saw-thooth function with $2^k$ \"theeth\" $\\big)$\n",
    "\n",
    "* \"on the fly\" add the scaled sawthooth functions in an additional node $\\Rightarrow$ precision $\\color{green}{\\mathcal{O}(2^{-k})}$ with ReLU networks of size $\\color{green}{\\mathcal{O}(k)}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 254
    },
    "id": "f0Xri3YnraWJ",
    "outputId": "3756085e-468e-41e7-886d-ea55b3eb5f19"
   },
   "outputs": [
    {
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    }
   ],
   "source": [
    "display_svg(SVG(\"figures/square_approx.svg\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 368,
     "referenced_widgets": [
      "3eb5c94b89f7482a86210be9dffbc2a7",
      "73e4bc7ef11e467c842fd54a77f87e58",
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      "651de4b2dcea45d184a6153b8dcce2f1"
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    },
    "id": "jrjTa8I7O1UD",
    "outputId": "6f0eaaa2-0cd7-4e4f-88f9-92373aeaaf0b"
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "2dace4019e41461eb8ac361f6a2e28fb",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=3, description='k', max=5, min=1), Output()), _dom_classes=('widget-inte…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "square = np.poly1d([1, 0, 0])\n",
    "interact(\n",
    "    lambda k: plot_1d(\n",
    "        [R(nn._square(k)), square],\n",
    "        titles=[\"approximation of the squaring function\"],\n",
    "        interval=[0, 1],\n",
    "    ),\n",
    "    k=(1, 5),\n",
    ");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ihXH0HO1K8UE"
   },
   "source": [
    "*...the same construction can be used when measuring the\n",
    "approximation error in the $W^{1,\\infty}$ norm. In particular, the depth and the number of weights and neurons\n",
    "of the network do not grow asymptotically faster to satisfy the approximation accuracy with respect to\n",
    "this stronger norm.* [[4]](https://arxiv.org/abs/1902.07896), [[8]](https://www.sam.math.ethz.ch/sam_reports/counter/ct.php?file=/sam_reports/reports_final/reports2019/2019-07_fp.pdf)\n",
    "\n",
    "Let $f(x)=x^2$. There are ReLU networks $\\Phi_\\varepsilon$ of size $\\color{green}{\\mathcal{O(\\log(1/\\varepsilon))}}$ satisfying $\\color{green}{\\|\\mathcal{R}\\Phi-f \\|_{W^{1,\\infty}((0,1))} \\le \\varepsilon}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "OCvM1DnDMOnc"
   },
   "source": [
    "---\n",
    "**Reminder:** \n",
    "\n",
    "$$ W^{n,p}(\\Omega):= \\{g\\in L^p(\\Omega): D^\\alpha g \\in L^p(\\Omega)  \\quad \\forall\\alpha\\in\\mathbb{N}_0^d \\quad \\text{with} \\quad |\\alpha|\\le n\\}$$\n",
    "\n",
    "$D^\\alpha g$ is the $\\alpha$-th weak derivatives of $g$, if $$\\int_{\\Omega} g (D^{\\alpha} \\phi) = (-1)^{|\\alpha|} \\int_{\\Omega} (D^{\\alpha} g) \\phi$$ for all $\\phi\\in C^\\infty_c(\\Omega)$.\n",
    "\n",
    "\n",
    "\n",
    "$$\\|g\\|_{W^{n,\\infty}(\\Omega)}:=\\max_{0\\le|\\alpha|\\le n} \\|D^{\\alpha} g\\|_{L^\\infty(\\Omega)}$$\n",
    "\n",
    "\n",
    "\n",
    "$\\big($ $\\Phi$ approximates $f$ in $W^{1,\\infty}$ $\\big)$ $\\Leftrightarrow$ $\\big($ $\\mathcal{R}(\\Phi)$ and its weak gradient uniformly approximate f and the weak gradient of f , respectively $\\big)$\n",
    "\n",
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "s-8Fv49VqdO0"
   },
   "source": [
    "Next we extent this approximation to the interval $[-B,B]$ (without altering the magnitude of its parameters or the asymptotic size scaling)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 70
    },
    "id": "vAYdpKxQrmlH",
    "outputId": "b72dcece-54eb-4085-f648-d4a33bfd01a5"
   },
   "outputs": [
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       "</g>\n",
       "</g>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SVG(\"figures/square.svg\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 368,
     "referenced_widgets": [
      "3c0839c6e46f4bdb864e2553fddbb876",
      "4f87fa1f882f4d66aae224615657a498",
      "c13d1fad332141439ecccdf1e6939c20",
      "cfaefa51c0f54d968ccbc3706f100dbc",
      "ea94f996271c4753a87c256accd97265",
      "6c29965ef81748e6b574e45086b89d5a",
      "292f43d787eb4d81963fccaf8e706ff3"
     ]
    },
    "id": "f2Y6Ir8wO-dN",
    "outputId": "896e0ec1-2a4e-470a-c364-6fc76c1a0db4"
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "2392f57afccb446986584d1fa1046617",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.55, description='eps', max=1.0, min=0.1), Output()), _dom_classes=('…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# visualization of squaring function approximation\n",
    "interact(\n",
    "    lambda eps: plot_1d(\n",
    "        [R(nn.square(eps)), square],\n",
    "        [D(nn.square(eps)), np.polyder(square)],\n",
    "        titles=[\n",
    "            \"approximation of the squaring function\",\n",
    "            \"approximation of the derivative\",\n",
    "        ],\n",
    "        interval=[-1, 1],\n",
    "        sharey=False,\n",
    "    ),\n",
    "    eps=(0.1, 1.0),\n",
    ");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 99,
     "referenced_widgets": [
      "faa6783fba204c3c8048d6123e40fe9e",
      "c892375551d24a198a9e2b548c5c77eb",
      "6d29b26228c94a2fb539bcc5c78ede7b",
      "7c3babebb317448cb3c0d3d7fadff733",
      "78616a90e8b349b3a764610120183dea",
      "f90141f67f8e428a9463b48b1db383a8",
      "9140094703854108b8f095b6970e1fc1",
      "7f9d48a73eb445f5b288281b2d9a7438",
      "7e0ccdebfc334501984160f6b8179419",
      "9cc04f9136ae4a2cb3ca169dfc2319a4"
     ]
    },
    "id": "8NQto8onO3kj",
    "outputId": "9c5ed6b4-a591-43de-9244-e6762530cc03"
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "ceec1d10af9e40f5ab97313fa5342aad",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.05, description='eps', max=0.1, min=0.01, step=0.01), FloatSlider(va…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# exploring the squaring function approximation networks\n",
    "interact(\n",
    "    lambda eps, bound: print(nn.square(eps, bound=bound).describe()),\n",
    "    eps=(0.01, 0.1, 0.01),\n",
    "    bound=(1.0, 100.0),\n",
    ");\n",
    "# note that the parameter magnitude is bounded (by leveraging the depth of the network)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 286
    },
    "id": "GfbE1A3AOuJ5",
    "outputId": "31c80c9f-f3c9-4f8e-c9af-3a6c28687435"
   },
   "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": [
    "# emprical test of logarithmic parameter scaling\n",
    "eps = 1 / np.linspace(1, 1e10, num=1000)\n",
    "y = [nn.square(e).num_parameters for e in eps]\n",
    "plt.semilogx(eps, y, linewidth=2)\n",
    "plt.xlabel(\"eps\")\n",
    "plt.ylabel(\"#Parameters\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ixM1QTuKOmd_"
   },
   "source": [
    "### Multiplication, Polynomials, and Smooth Functions\n",
    "\n",
    "Combing (composition, linear combination) the squaring networks (and exactly representable easy networks) one gets \n",
    "* networks approximating multiplication: $\\color{green}{xy=\\left|\\frac{x+y}{2}\\right|^2-\\left|\\frac{x-y}{2}\\right|^2}$\n",
    "* polynomials \n",
    "* functions whose first $n\\geq 1$ (weak) derivatives are bounded: local Taylor approximation\n",
    "\n"
   ]
  },
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     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "SVG(\"figures/mult.svg\")"
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  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 99,
     "referenced_widgets": [
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    "id": "W1mh3M3gqdPP",
    "outputId": "2b0705a1-bb17-4ea3-dd89-8385bc9bf93f"
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "4f0b75df61a34460bb64d24cc1cb3b57",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.0, description='x', max=1.5, min=-1.5), FloatSlider(value=0.0, descr…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# exploring the networks approximating the multiplication function (on [-1,1]!)\n",
    "interact(\n",
    "    lambda x, y: print(\n",
    "        f\"Realization of multiplication network at ({x},{y}): {float(R(nn.mult(1e-2, bound=1.))([[x,y]]))}\"\n",
    "    ),\n",
    "    x=(-1.5, 1.5),\n",
    "    y=(-1.5, 1.5),\n",
    ");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 304
    },
    "id": "zS4D7YoQr_YU",
    "outputId": "5f4f1663-ed57-48c4-c267-3718f87ad8e3"
   },
   "outputs": [
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       "</svg>"
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      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SVG(\"figures/monomials.svg\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 99,
     "referenced_widgets": [
      "a5beb92f90a84236996758483a9b8ef4",
      "168a502eeee748cfb866f42d03db065e",
      "36d321a3582f47d3a215293539d35f00",
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      "1f6db14b5f66475e8f665ace39fcd66c",
      "975b5b45cea04bf291d152d47683fa3d"
     ]
    },
    "id": "P94YB27lAfNY",
    "outputId": "bb48b8f7-577c-4d08-a1bf-4bb97bc5b3e6"
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "70ff4f41d5a1401fa13aacad528afe6f",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.0, description='x', max=10.0, min=-10.0), Checkbox(value=True, descr…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Testing the output\n",
    "interact(\n",
    "    lambda x, const: print(\n",
    "        f\"Realization of monomial network at {x}: {R(nn.monomial(3, 0.1, bound=10., const=const))([[x]]).numpy()[0]}\"\n",
    "    ),\n",
    "    const=True,\n",
    "    x=(-10.0, 10.0),\n",
    ");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 336
    },
    "id": "1sjCZHDDmmBX",
    "outputId": "0ee81511-f8c9-4275-b831-481460c2e259"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# visualization of polynomial approximation\n",
    "p = [-2.0, 2.0, 1.77, 1.0]\n",
    "poly = np.poly1d(p)\n",
    "pa = nn.poly(p, eps=1.0)  # as this are worst case estimates, eps can be rather large\n",
    "plot_1d(\n",
    "    [R(pa), poly],\n",
    "    [D(pa), np.polyder(poly)],\n",
    "    titles=[f\"approximation of the polynomial\", \"approximation of the derivative\"],\n",
    "    interval=[-2, 2],\n",
    "    sharey=False,\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 35
    },
    "id": "Sy19hzwWzDl9",
    "outputId": "a820dced-7e0b-4859-8f7f-41b2922ef951"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Depth: 34, Width: 16, Magnitude: 4.0, #Parameters: 3325, Connectivity: 771'"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pa.describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "7qWOuKeWCBt0"
   },
   "source": [
    "### Theoretical Results \n",
    "\n",
    "**...for classes of Sobolev-regular functions:** [[4]](https://arxiv.org/abs/1902.07896)\n",
    "\n",
    "There exists $c=c(d,n,B)> 0$ such that for every $\\varepsilon > 0$ and every\n",
    "$$f\\in\\{g\\in W^{n,\\infty}((0,1)^d): \\|g\\|_{W^{n,\\infty}((0,1)^d)}\\le B\\}$$\n",
    "there exists a ReLU network $\\Phi$ with\n",
    "$$\\|\\mathcal{R}\\Phi-f \\|_{W^{1,\\infty}((0,1)^d)} \\le \\varepsilon$$\n",
    "and\n",
    "$$Depth(\\Phi)\\le c \\cdot log(\\varepsilon^{-n/(n-1)}), \\quad Connectivity(\\Phi)\\le  c \\cdot \\varepsilon^{-d/(n-1)} \\cdot log^2(\\varepsilon^{-n/(n-1)}).$$\n",
    "\n",
    "Corresponding results for:\n",
    "\n",
    "* fractional Sobolev spaces $W^{s,p}$ with $0\\le s \\le 1$ (Banach space interpolation argument) and $1\\le p \\le \\infty$\n",
    "* any continuous piecewise linear activation function with $1\\le M < \\infty$ breakpoints\n",
    "* lower bound: $Connectivity(\\Phi)\\ge  c \\cdot \\varepsilon^{-d/2(n-1)}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "id": "64feM4qoqdPo"
   },
   "outputs": [],
   "source": [
    "# You can try to construct this networks (perhaps for d=1) using our framework\n",
    "# 1. construct partition of unity\n",
    "# 2. predefine derivatives of f (or use automatic differentiation frameworks)\n",
    "# 3. use the neural network approximating polynomials\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "WKMPuWlVs6xT"
   },
   "source": [
    "## Additional Material"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "K__ooWrF72xb"
   },
   "source": [
    "### ReLU derivative\n",
    "\n",
    "**Problem: Classical Chain-Rule fails!**\n",
    "\n",
    "...although all ReLU networks are a.e. differentiable.\n",
    "\n",
    "Very simple counterexample: \n",
    "\n",
    "$u(x):=\\operatorname{ReLU}(x)$\n",
    "\n",
    "$v(x):=0$ \n",
    "\n",
    "formally apply the chain rule: $$D(u\\circ v)(x)=\\color{blue}{D u(v(x))}\\cdot D v(x)$$\n",
    "\n",
    "$D u(y)$ is well-defined for all $y\\neq 0$, but the expression $\\color{blue}{D u(v(x))}$ is not defined for any $x\\in\\mathbb{R}$.\n",
    "\n",
    "\n",
    "**Inner function maps a set of positive measure into a set where the derivative of the outer function does not exist.**\n",
    "\n",
    "\n",
    "**What happens in PyTorch / Tensorflow when computing the gradients in the back-propagation (as this is based on the chain-rule)??**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 336
    },
    "id": "xO7sMgj933fu",
    "outputId": "07a2b211-2e41-4ac8-ca8d-a271734e2d3a"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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gmJLq2JOvv01P7yEmj+3g8x+5IOtypIY28Cqfz1lUzDAl1amUEncUFjS+9WMzGdXqw1mSsuDoK9WpNT27eWXLXsaNbufLc6dmXY7U8FLhqp12plTMMCXVqeWFrtRN13TT0d6acTWS1LwMU1Idem37Pla92Utneys3XD0963KkpnB8zpTv51MRw5RUhwa6UkvmTWNsZ3vG1UhSczNMSXVm8+53+d6r2xnVEixdMCPrcqSmMbDQsXOmVMwwJdWZu1a/RS7BdZdPZtLYjqzLkaSmZ5iS6kjvgT4eW7sVgGULZ2ZcjdRcXE1GpRimpDpy37Mb6OvPce0lE5k1cUzW5UiSMExJdePA4aM8uGYT4ILGUhbev86Uk6Z0IsOUVCceemEz+w/3M797HFdN78q6HElSgWFKqgN9/ce495kNgF0pKSvOmVIphimpDjz+0jZ27u9jzvljWDR7fNblSE3NF/lUzDAl1bhcLnHnqvxFOm9bONP5GlJWbE2pBMOUVON+8PpOenoPMXlsB5//yKSsy5Gans9nVMwwJdWwlBJ3PP0WALcsmEFbqw9ZKSvJ1pRKcGSWathzPXt4Zctexo1u5yvzpmVdjiRc6FgfZJiSathAV+rGq7vpaG/NuBqpuSUbUyrBMCXVqNe272PVm710trdyw9XTsy5HUoFzplTMMCXVqOVP59/Bt2TeNLpGt2dcjSQ7UyrFMCXVoM273+V7r25nVEuwdMGMrMuRNIiNKRUzTEk16K7Vb5FLcN3lk5k0tiPrciThu/lUmmFKqjG7Dvbx2NqtACxbODPjaiQVc86UihmmpBpz30820tef49pLJjJr4pisy5EkDcEwJdWQg339PLBmIwC3L7IrJdWS9yeg25rSiQxTUg156PnN7D/cz/zucVw1fVzW5UiSymCYkmpEX/8x7nkmfzmEZXalpJoz0JhyzpSKGaakGvHES9vZub+P2RPH8PHZE7IuR5JUJsOUVANyucTyVfmlY5Ytmkn41FeqOQNzpnx0qlhZYSoiPh0Rb0TE+oj4Wol9vhwRr0fEaxHxreqWKTW2H7y+k57eQ0we28HnPzIp63IkSadh1FA7REQr8E3gU8BW4MWIWJFSen3QPrOAfwH8WkrplxHhaxRSmVJKxxc0vmXBDNpabRhLtcmLdurkyhm15wPrU0o9KaUjwMPAdUX73AJ8M6X0S4CU0jvVLVNqXM/17OGVLXvp6mzjy/OmZl2OpCH4MryKlROmJgNbBn2+tbBtsIuBiyPiJxHxXER8uloFSo1ueaErddM1M+hsH7JZLCkjLnSsUqo1co8CZgGLgCnAqoj4cEpp7+CdIuJW4FaAadOmVelbS/Xrte37ePrNXjraWrnh6ulZl6NTcPzSAPtSKlZOZ2obMPi1hymFbYNtBVaklI6mlDYAb5IPVydIKd2VUpqbUpo7fvz4M61Zahh3Pp2/rtT186fRNbo942p0Ko5fsjGlUsoJUy8CsyJiRkS0A0uAFUX7PE6+K0VEnEf+Zb+e6pUpNZ7Nu9/lu69uZ1RLsHTBjKzLkVQmp0yp2JBhKqXUD3wV+D6wDng0pfRaRHwjIhYXdvs+sDsiXgd+DPxeSmn3cBUtNYK7V/eQS3Dd5ZOZNLYj63IkDSE5aUollDVnKqW0ElhZtO3rg24n4J8WPiQNYdfBPh5dm39fx7KFLh0j1RM7UyrmBW2kDNz3k4309ee49pKJzJo4JutyJJXBvpRKMUxJI+xgXz8PrNkIwO0uaCzVnfD9fCpimJJG2EPPb2b/4X7mdXdx1fRxWZcjqUxOmVIphilpBPX1H+OeZ/JvdL190YUZVyPpjNiYUhHDlDSCnnhpOzv39zF74hg+PtslLKV6YmNKpRimpBGSyyWWr8ovHbNs0UzX95LqlI9cFTNMSSPkB6/vpKf3EJPHdvD5j0zKuhxJp8nrTKkUw5Q0AlJK3FFY0PiWBTNoa/WhJ9Uru8oq5ogujYDnevbwypa9dHW28eV5U4f+AklS3TBMSSNgeaErddM1M+hsL2vhAUk1yr6UihmmpGH22vZ9PP1mLx1trdxw9fSsy5EkVZlhShpmdz6dv67UkvlT6RrdnnE1ks7UwPxzp0ypmGFKGkabd7/Ld1/dzqiWYOkCl46RpEZkmJKG0d2re8glWHz5JCaP7ci6HEkVSF62UyUYpqRhsutgH4+u3QLAsoUuHSM1Chc6VjHDlDRM7vvJRvr6c1x7yQQunjgm63IkVchrdqoUw5Q0DA729fPAmo2ACxpLjcYJ6CpmmJKGwUPPb2b/4X7mdXdx1fRxWZcjqQrsTKkUw5RUZX39x7jnmfzlEOxKSY3HxpSKGaakKnvipe3s3N/H7Ilj+PjsCVmXI6lKfDefSjFMSVWUyyWWr8ovHXPbwpkuiCo1Ih/WKmKYkqroyXU76ek9xOSxHXzhsklZlyOpipwzpVIMU1KVpJS446l8V2rpghm0tfrwkhqR15lSMUd7qUqe37CHl7fspauzja/Mm5p1OZKqzMaUSjFMSVUy0JW68ZpuOttHZVyNpOHiVEgVM0xJVfD69v08/WYvHW2t3Hh1d9blSBoOtqZUgmFKqoLlT+e7UkvmT6VrdHvG1UgaTjamVMwwJVVo8+53+e6r2xnVEixdMDPrciQNE68zpVIMU1KF7l7dQy7B4ssnMXlsR9blSBpmXj9OxQxTUgV2Hezj0bVbAFi20KVjpEbmdaZUimFKqsB9P9lIX3+Oay+ZwMUTx2RdjqQRYGNKxQxT0hk62NfPA2s2AnalpGZgY0qlGKakM/TwC5vZf7ifed1dzO0el3U5kkaIjSkVM0xJZ+BIf457Vm8A7EpJzSI5aUolGKakM/D4y9t4e/9hZk8cw8dnT8i6HElShgxT0mnK5dLxi3TetnAmLS02/aVm4gR0FTNMSafpyXU76ek9xOSxHXzhsklZlyNphPgin0oxTEmnIaV0fEHjpQtm0NbqQ0hqPramdCL/Ekin4fkNe3h5y166Otv4yrypWZcjaQQ5/1ylGKak0zDQlbrxmm4620dlXI2kLDhnSsUMU1KZXt++n6ff7KWjrZUbr+7OuhxJI8zGlEopK0xFxKcj4o2IWB8RXzvFfr8RESki5lavRKk23Lkq35VaMn8qXaPbM65GUlZsTKnYkGEqIlqBbwKfAS4Fro+IS0+y3xjg/wKer3aRUta27HmX77yynVEtwdIFM7MuR1IWnDSlEsrpTM0H1qeUelJKR4CHgetOst+/Bv4YOFzF+qSacPfqHnIJFl8+icljO7IuR1KGwklTKlJOmJoMbBn0+dbCtuMi4kpgakrpe1WsTaoJuw728ciL+YeAS8dIzcu+lEqpeAJ6RLQA/wn43TL2vTUi1kbE2t7e3kq/tTQi7n92I339Oa69ZAIXTxyTdTnKiOOXBtiXUrFywtQ2YPAFdaYUtg0YA/xd4KmI2Ah8FFhxsknoKaW7UkpzU0pzx48ff+ZVSyPkYF8/9z+7EbAr1ewcv+SUKZVSTph6EZgVETMioh1YAqwYuDOltC+ldF5KqTul1A08ByxOKa0dloqlEfTwC5vZf7ifudO7mNs9LutyJNUAp0yp2JBhKqXUD3wV+D6wDng0pfRaRHwjIhYPd4FSVo7057hn9QYAbl9kV0pqdslZUyqhrEs4p5RWAiuLtn29xL6LKi9Lyt7jL2/j7f2HuXjiOXx89oSsy5FUI8JZUyriFdClk8jlEnc+nb9I57KFF9LS4uApNTvnTKkUw5R0Ek+u28lbvYeYPLaDL1w2KetyJNUQ50ypmGFKKpJSOr6g8dIFM2hr9WEiyc6USvOvhFTk+Q17eHnLXro62/jKvKlDf4EkqakZpqQiywtzpW68ppvO9rLeoyGpCdiYUimGKWmQ17fv56k3euloa+XGq7uzLkeSVAcMU9Igd67Kd6W+Mm8qXaPbM65GUi1JhUlTTkBXMcOUVLBlz7t855XtjGoJli6YkXU5kqQ6YZiSCu5e3UMuweLLJjGlqzPrciTVKC/aqWKGKQnYdbCPR17cAsBtLmgsSToNhikJuP/ZjfT15/jknAnMPn9M1uVIqmHOmVIxw5Sa3sG+fh5YswlwQWNJpXnRTpVimFLTe/iFzex77yhzp3cxt3tc1uVIqnF2plTMMKWmdqQ/xz2rNwB2pSSdWvKynSrBMKWm9vjL23h7/2EunngOH589IetyJNUB382nYoYpNa1cLnFnYemY2z52IS0tDpCSSnPOlEoxTKlpPbluJ2/1HmLSuWez+PJJWZcjqU44Z0rFDFNqSiml4wsaL10wk7ZWHwqSTs3GlErxL4ia0gsb9vDS5r2M7WxjyfypWZcjqY7YmFIxw5Sa0h2FrtSNV3fT2T4q42ok1YPkpCmVYJhS01m3Yz9PvdFLR1srN17TnXU5kuqMc6ZUzDClpjMwV+or86YybnR7xtVIqhf2pVSKYUpNZcued/nuqzsY1RIsXTAj63Ik1SVbUzqRYUpN5e7VPRzLJRZfNokpXZ1ZlyOpntiaUgmGKTWNXQf7eOTFLQDcttClYyRJ1WGYUtO4/9mN9PXn+OScCcw+f0zW5UiqMwONKSegq5hhSk3hUF8/D6zZBMAyFzSWJFWRYUpN4aEXNrPvvaPMnd7FvO5xWZcjqQ4NXGfKxpSKGabU8I7057hn9QYAljlXSpJUZYYpNbwnXt7G2/sPc/HEc/jEnAlZlyOpTr0/Z8relE5kmFJDy+XeX9D4to9dSEuLg6AkqboMU2poP1y3k7d6DzHp3LNZfPmkrMuRVMcGlubzKZmKGabUsFJKxxc0XrpgJm2t/rpLkqrPvy5qWC9s2MNLm/cytrONJfOnZl2OpAbhlCkVM0ypYQ10pW68upvO9lEZVyOp3iXXk1EJhik1pHU79vPUG72c3dbCjdd0Z12OpAYSzppSEcOUGtLAO/iWzJvGuNHtGVcjqREkG1MqwTClhrNlz7t899UdtLYESxfMyLocSQ3GOVMqZphSw7lndQ/HconrLpvElK7OrMuR1CDsTKkUw5Qayu6DfTyydgsAt7l0jCRpBJQVpiLi0xHxRkSsj4ivneT+fxoRr0fEqxHx1xExvfqlSkO7/9mNHD6a45NzJjD7/DFZlyNJagJDhqmIaAW+CXwGuBS4PiIuLdrtJWBuSukjwLeBP6l2odJQDvX1c/+aTQAsW2RXStLwcM6UipXTmZoPrE8p9aSUjgAPA9cN3iGl9OOU0ruFT58DplS3TGloD72wmX3vHeWq6V3M6x6XdTmSGkxy0pRKKCdMTQa2DPp8a2FbKTcDf1VJUdLpOtKf457VGwC43blSkoZR2JpSkapeFjoi/hEwF1hY4v5bgVsBpk2bVs1vrSb3xMvbeHv/YWZNOIdPzJmQdTlqQI5fsi+lUsrpTG0DBi9sNqWw7QQRcS3w+8DilFLfyQ6UUrorpTQ3pTR3/PjxZ1Kv9AG5XDp+kc5lCy+kpcVnjao+xy9JpZQTpl4EZkXEjIhoB5YAKwbvEBFXAHeSD1LvVL9MqbQfrtvJW72HmHTu2Sy+fFLW5UhqUANTpny6pmJDhqmUUj/wVeD7wDrg0ZTSaxHxjYhYXNjtPwDnAI9FxMsRsaLE4aSqSikdX9B46YKZtLV66TRJ0sgqa85USmklsLJo29cH3b62ynVJZXlhwx5e2ryXsZ1tLJk/degvkKQzlAqzppx/rmI+jVddG5grdePV3XS2V/X9FJIklcUwpbq1bsd+fvxGL2e3tXDjNd1ZlyOpwb0/Z8rWlE5kmFLdurPQlVoybxrjRrdnXI0kqVkZplSXtux5l++8uoPWlmDpghlZlyOpCQxcZ8o5UypmmFJdumd1D8dyicWXTWJKV2fW5UiSmphhSnVn98E+HlmbX+HotoUzM65GUrPwOlMqxTClunP/sxs5fDTHJ+ZMYM75H8q6HElSkzNMqa4c6uvn/jWbALh9kQsaSxpJtqZ0coYp1ZWHXtjMvveOctX0LuZ1j8u6HEmSDFOqH0f6c9z7zAYAbl9oV0pSNrzOlIoZplQ3nnh5Gzv2HWbWhHP4xJwJWZcjqckMTECXihmmVBdyucSdq3oAuG3hhbS0+MxQUja8zpSKGaZUF364bifr3znIpHPPZvFlk7IuR1ITsjGlUgxTqnkpJe4oLB1z84KZtI/y11ZSdmxMqZh/lVTzXtiwh5c272VsZxtL5k3NuhxJTSo5aUolGKZU85YXulI3XN3N6LNGZVyNJEknMkyppq3bsZ8fv9HL2W0t3HRNd9blSGpi7y907At9OpFhSjXtzkJXasm8aYwb3Z5xNZIkfZBhSjVry553+c6rO2htCZYumJF1OZKanAsdqxTDlGrWPat7OJZLLL5sElO6OrMuR5KkkzJMqSbtPtjHI2u3AHDbwpkZVyNJ77+bzylTKmaYUk26/9mNHD6a4xNzJjDn/A9lXY4kSSUZplRzDvX1c/+aTQAsc0FjSTXi+Lv5nDWlIoYp1ZyHXtjMvveOctX0LuZ1d2VdjiRJp2SYUk050p/j3mc2APmulNdzkVQz3m9NSScwTKmmrHhlOzv2HWbWhHP45JwJWZcjSdKQDFOqGblcOr50zG0LL6Slxad/kmrH+1dAz7QM1SDDlGrGX//tO6x/5yCTzj2bxZdNyrocSZLKYphSTUgpccdT6wG4ecFM2kf5qympthy/zlTGdaj2+BdLNeHFjb/kbzbv5dyONpbMm5p1OZIklc0wpZow0JW68ZpuRp81KuNqJOmD3p8zZW9KJzJMKXPrduznx2/0cnZbCzdd0511OZIknRbDlDJ3Z+EdfEvmTWPc6PaMq5GkU7MvpWKGKWVqy553+c6rO2htCW7+9RlZlyNJJaU09D5qToYpZereZzZwLJdYfNkkpo7rzLocSRqSU6ZUzDClzOw+2MfDL24G4LaFMzOuRpJOLWFrSidnmFJm7l+zicNHc3xizgTmnP+hrMuRJOmMGKaUiUN9/dz/7EYgv6CxJNW6gTlT4RR0FTFMKRMPv7iFfe8d5cppY5nX3ZV1OZIknTHDlEbckf4c96zuAeD2RRd5ATxJdcGFjlWKYUojbsUr29mx7zCzJpzDJ+dMyLocSZIqUlaYiohPR8QbEbE+Ir52kvvPiohHCvc/HxHdVa9UDSGXSywvXKTztoUX0tLiUzxJdcI386mEIcNURLQC3wQ+A1wKXB8RlxbtdjPwy5TSRcB/Bv642oWqMfz1377D+ncOcsG5Z7P4sklZlyNJUsXKWVF2PrA+pdQDEBEPA9cBrw/a5zrgDwu3vw38eURESpVfL/anm37Jyp/tqPQwqgEpwRMvbwNg6YKZtI/yVWY1tqPHcvz7v/rbrMtQlfTncoBzpvRB5YSpycCWQZ9vBX611D4ppf6I2Af8CrBr8E4RcStwK8C0adPKKvDNnQe495kNZe2r+jClq4Ml86ZmXYZ0Ws5k/DqWS45fDaazvZVW05SKlBOmqialdBdwF8DcuXPL6lpdMW0sf/C5S4a1Lo2cttYWPvPh8xl91oj+6kkVO5Pxa1RLOH41mMunjmVUq111naicv2jbgMFthCmFbSfbZ2tEjALOBXZXo8A553/Iq2NLqkujWltYusClkqRGV068fhGYFREzIqIdWAKsKNpnBXBj4fZvAj+qxnwpSZKkWjdkZ6owB+qrwPeBVuAvUkqvRcQ3gLUppRXAvcCDEbEe2EM+cEmSJDW8siaupJRWAiuLtn190O3DwJeqW5okSVLtcxadJElSBQxTkiRJFTBMSZIkVcAwJUmSVAHDlCRJUgUMU5IkSRUwTEmSJFUgsrpQeUT0AptO40vOo2jh5AbUDOcInmcjOd1znJ5SGj9cxYwUx6+SmuE8m+EcwfM8mZLjV2Zh6nRFxNqU0tys6xhOzXCO4Hk2kmY4x2polv+nZjjPZjhH8DxPly/zSZIkVcAwJUmSVIF6ClN3ZV3ACGiGcwTPs5E0wzlWQ7P8PzXDeTbDOYLneVrqZs6UJElSLaqnzpQkSVLNqdkwFRFfiojXIiIXESVn2kfEpyPijYhYHxFfG8kaKxUR4yLiyYj4ReHfrhL7HYuIlwsfK0a6zjM11M8mIs6KiEcK9z8fEd0ZlFmRMs7xpojoHfTzW5pFnZWIiL+IiHci4ucl7o+I+LPC/8GrEXHlSNdYa5ph/ILGHsOaYfwCx7DC/ZWPYSmlmvwALgFmA08Bc0vs0wq8BcwE2oFXgEuzrv00zvFPgK8Vbn8N+OMS+x3MutYzOLchfzbAPwaWF24vAR7Juu5hOMebgD/PutYKz/NjwJXAz0vc/1ngr4AAPgo8n3XNWX80w/hVOIeGHMOaYfw6jfN0DCvjo2Y7UymldSmlN4bYbT6wPqXUk1I6AjwMXDf81VXNdcD9hdv3A1/MrpSqK+dnM/j8vw18MiJiBGusVL3//pUlpbQK2HOKXa4DHkh5zwFjI+KCkamuNjXJ+AWNO4Y1w/gFjfE7OKSRGMNqNkyVaTKwZdDnWwvb6sXElNKOwu23gYkl9js7ItZGxHMR8cWRKa1i5fxsju+TUuoH9gG/MiLVVUe5v3+/UWgdfzsipo5MaSOq3h+HWWmE/7dGHcOaYfwCx7ABFT8WR1W1nNMUET8Ezj/JXb+fUnpipOsZDqc6x8GfpJRSRJR6a+X0lNK2iJgJ/CgifpZSeqvatWpYfAd4KKXUFxG3kX8m+4mMa1IVNMP4BY5hcgwrR6ZhKqV0bYWH2AYMTslTCttqxqnOMSJ2RsQFKaUdhZbiOyWOsa3wb09EPAVcQf517lpWzs9mYJ+tETEKOBfYPTLlVcWQ55hSGnw+95CfY9Joav5xOByaYfyCph3DmmH8AsewARU/Fuv9Zb4XgVkRMSMi2slPAqyLd4oUrABuLNy+EfjAs9mI6IqIswq3zwN+DXh9xCo8c+X8bAaf/28CP0qF2YB1YshzLHrdfTGwbgTrGykrgBsK74j5KLBv0Es/Kq3exy9o3DGsGcYvcAwbUPkYlvUs+1PMvv/fyL9u2QfsBL5f2D4JWFk0C/9N8s9yfj/ruk/zHH8F+GvgF8APgXGF7XOBewq3rwF+Rv5dFj8Dbs667tM4vw/8bIBvAIsLt88GHgPWAy8AM7OueRjO8Y+A1wo/vx8Dc7Ku+QzO8SFgB3C08Ji8GVgGLCvcH8A3C/8HP6PEu9ea6aMZxq9C/Q07hjXD+FXmeTqGlfHhFdAlSZIqUO8v80mSJGXKMCVJklQBw5QkSVIFDFOSJEkVMExJkiRVwDAlSZJUAcOUJElSBQxTkiRJFfj/AZDs7kJuEMEwAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 720x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "act = P(\n",
    "    [(np.array([[1.0]]), np.array([0.0])), (np.array([[1.0]]), np.array([0.0]))],\n",
    "    name=\"act\",\n",
    ")\n",
    "plot_1d(R(act), D(act), titles=[\"ReLU\", \"derivative\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "Sd3Xh4X7o6xl",
    "outputId": "7cf3c6d4-cd2a-48c4-ce17-24aa819755df"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "float(D(act)([[0.0]]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "XrmWYrBVJerG"
   },
   "source": [
    "The derivative of the ReLU at zero is set to zero in PyTorch and Tensorflow, i.e.\n",
    "$\\color{blue}{\\frac{\\partial \\operatorname{ReLU}}{\\partial x}(x) :=1_{(0,\\infty)}(x)}$.\n",
    "\n",
    "Note that the chain rule holds by definition everywhere, i.e. \n",
    "\n",
    " $$\\color{green}{\\mathcal{D}(\\Psi\\circ\\Phi)=\\mathcal{D}\\Psi(\\mathcal{R}\\Phi)\\cdot\\mathcal{D}\\Phi}$$\n",
    "\n",
    "and, furthermore, $\\color{green}{\\mathcal{D}\\Phi=D\\mathcal{R}\\Phi \\quad a.e.}$\n",
    "\n",
    "\n",
    "\n",
    "**However:** On the nullset the derivative can be arbitrary wrong, e.g. take the network $\\Phi=\\left(\\left( (1 , -1 ),(0 ,0)\\right),\\left( \\left(\\begin{matrix} a \\\\-b \\end{matrix}\\right),0\\right)\\right)$ then it holds:\n",
    "\n",
    "1. $\\mathcal{R}\\Phi(x)=a\\operatorname{ReLU}(x)-b\\operatorname{ReLU}(-x)=\\begin{cases}  bx, &x< 0 \\\\  ax, &x\\ge 0 \\end{cases}$\n",
    "2. $ \\mathcal{D}\\Phi(x)=\\begin{cases}  b, &x< 0 \\\\  a, & x> 0 \\\\ \\color{blue}0, & x=0\\end{cases}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "cellView": "both",
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 418,
     "referenced_widgets": [
      "070b3f91903c4c2f8664b2147c55b130",
      "b8abf7ee2aa5446f90a95ca484896f62",
      "2270bcbec47947d08c265c9d46beebc8",
      "27d0953c4f0b4e7caaeed696b20e4459",
      "6d5598ef1c1746309ea3cdcced37d4c1",
      "4391a0958cf448b9a1cf6f317101ee2f",
      "30c91d461eab497288431c91d65f0409",
      "2ec0d0a286a144568a8ae565b602d574",
      "9ba1621ea6924b12b0f41d1b58e39fe7",
      "3d4bf3c4f4824b4f8c4053eb5196e226"
     ]
    },
    "id": "akk8hLyBNPjZ",
    "outputId": "1e1e818a-2165-48df-8205-32f11dc08e76"
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "73f7b6624f094714bce890ab8650f4a0",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=1.5, description='a', max=2.0, min=1.0), FloatSlider(value=1.5, descri…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_print(a, b):\n",
    "    pa = P(\n",
    "        [\n",
    "            (np.array([[1.0, -1.0]]), np.array([0, 0])),\n",
    "            (np.array([[a], [-b]]), np.array([0.0])),\n",
    "        ]\n",
    "    )\n",
    "    plot_1d(\n",
    "        R(pa), D(pa), sharey=False, titles=[f\"{a}ReLU(x) - {b}ReLU(x)\", \"derivative\"]\n",
    "    )\n",
    "    print(f\"Derivative at x=0: {float(D(pa)([[0.]]))}\")\n",
    "\n",
    "\n",
    "interact(plot_print, a=(1.0, 2.0), b=(1.0, 2.0));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "duQFOOZlTboN"
   },
   "source": [
    "For convex $\\mathcal{R}\\Phi$ the values of $\\mathcal{D}\\Phi$ on the nullset do not necessarily lie in the respective subdifferentials of $\\mathcal{R}\\Phi$.\n",
    "\n",
    "The next figure illustrates this by showing the behavior of the (backpropagated) derivative of the neural network approximating the squaring function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 486
    },
    "id": "QiTl1lX5s7M6",
    "outputId": "2656b8d0-7060-48cf-81a7-c25de57051b4"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<IPython.core.display.Image object>"
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     },
     "execution_count": 37,
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    }
   ],
   "source": [
    "Image(\"figures/squaring_der.png\")"
   ]
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   "source": [
    "Although our results hold regardless of which value is chosen for the derivative of $\\operatorname{ReLU}$ at the origin, no choice will guarantee that all values of $\\mathcal{D} \\Phi$ lie in the respective subdifferentials of $\\mathcal{R}\\Phi$ (e.g., additionally consider $x \\mapsto ReLU(-ReLU(x))$)"
   ]
  },
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    "id": "O3LiTJ0ouCLt"
   },
   "source": [
    "### Global (Pointwise) Error Estimates\n",
    "\n",
    "For suitable functions $f$ with at most polynomially growing derivative we want ReLU networks $\\Phi_\\varepsilon$ with:\n",
    "\n",
    "\n",
    "$$ \n",
    "|f(x)-\\mathcal{R}{\\Phi_\\varepsilon}(x)|\\le \\varepsilon(1+\\|x\\|^{\\kappa}_2) \\quad \\color{blue}{x\\in\\mathbb{R}^d}\n",
    "$$\n",
    "$$\n",
    "    \\|D f(x) -\\mathcal{D}\\Phi_\\varepsilon(x)\\|_2 \\le C\\varepsilon^r(1+\\|x\\|_2^{\\kappa}) \\quad \\color{blue}{a.e. \\ x\\in\\mathbb{R}^d}\n",
    "$$\n",
    "\n",
    "Idea:\n",
    "1. ReLU networks $\\Phi_{\\varepsilon,B}$ with $\\|\\mathcal{R}\\Phi-f \\|_{W^{1,\\infty}((-B,B)^d)} \\le \\varepsilon$\n",
    "2.  ReLU network $\\Phi_B^{\\operatorname{char}}$ which represents the $d$-dimensional approximate characteristic function of $[-B,B]^d$ [[3]](https://arxiv.org/abs/1809.07669)\n",
    "3. ReLU network $\\Phi_{\\varepsilon,b}^{\\operatorname{mult}}$ approximating the multiplication function on $[-b,b]^2$ with error $\\varepsilon$\n",
    "4. Define the global approximation networks $\\Phi_\\varepsilon$ as the composition of $\\Phi_{\\varepsilon/2,b_\\varepsilon}^{\\operatorname{mult}}$ with the parallelization of $\\Phi_{B_\\varepsilon}^{\\operatorname{char}}$ and $\\Phi_{\\varepsilon/2,B_\\varepsilon+1}$ for suitable \n",
    "\n",
    "\\begin{equation}\n",
    "    B_\\varepsilon\\in \\mathcal{O}(\\varepsilon^{-1}) \\quad \\text{and} \\quad b_\\varepsilon\\in \\mathcal{O}(\\varepsilon^{-\\nu}).\n",
    "\\end{equation}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "id": "ebtkAgUpqdP8"
   },
   "outputs": [],
   "source": [
    "# You can also try to construct this networks (perhaps for d=1) using our framework\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#\n",
    "#"
   ]
  }
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      "top": null,
      "visibility": null,
      "width": null
     }
    },
    "1f47c49a4904400baa28984228557ab9": {
     "model_module": "@jupyter-widgets/output",
     "model_name": "OutputModel",
     "state": {
      "_dom_classes": [],
      "_model_module": "@jupyter-widgets/output",
      "_model_module_version": "1.0.0",
      "_model_name": "OutputModel",
      "_view_count": null,
      "_view_module": "@jupyter-widgets/output",
      "_view_module_version": "1.0.0",
      "_view_name": "OutputView",
      "layout": "IPY_MODEL_651de4b2dcea45d184a6153b8dcce2f1",
      "msg_id": "",
      "outputs": [
       {
        "image/png": 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\n",
        "metadata": {
         "needs_background": "light",
         "tags": []
        },
        "output_type": "display_data",
        "text/plain": "<Figure size 720x360 with 1 Axes>"
       }
      ]
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      "grid_template_areas": null,
      "grid_template_columns": null,
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      "overflow_x": null,
      "overflow_y": null,
      "padding": null,
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      "top": null,
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     }
    },
    "6d29b26228c94a2fb539bcc5c78ede7b": {
     "model_module": "@jupyter-widgets/controls",
     "model_name": "FloatSliderModel",
     "state": {
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      "_model_module": "@jupyter-widgets/controls",
      "_model_module_version": "1.5.0",
      "_model_name": "FloatSliderModel",
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      "_view_module": "@jupyter-widgets/controls",
      "_view_module_version": "1.5.0",
      "_view_name": "FloatSliderView",
      "continuous_update": true,
      "description": "eps",
      "description_tooltip": null,
      "disabled": false,
      "layout": "IPY_MODEL_9140094703854108b8f095b6970e1fc1",
      "max": 0.1,
      "min": 0.01,
      "orientation": "horizontal",
      "readout": true,
      "readout_format": ".2f",
      "step": 0.01,
      "style": "IPY_MODEL_f90141f67f8e428a9463b48b1db383a8",
      "value": 0.05
     }
    },
    "6d5598ef1c1746309ea3cdcced37d4c1": {
     "model_module": "@jupyter-widgets/output",
     "model_name": "OutputModel",
     "state": {
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      "_model_module": "@jupyter-widgets/output",
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      "_view_module": "@jupyter-widgets/output",
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      "_view_name": "OutputView",
      "layout": "IPY_MODEL_3d4bf3c4f4824b4f8c4053eb5196e226",
      "msg_id": "",
      "outputs": [
       {
        "image/png": 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\n",
        "metadata": {
         "needs_background": "light",
         "tags": []
        },
        "output_type": "display_data",
        "text/plain": "<Figure size 720x360 with 2 Axes>"
       },
       {
        "metadata": {
         "tags": []
        },
        "output_type": "stream",
        "stream": "stdout",
        "text": "Derivative at x=0: 0.0\n"
       }
      ]
     }
    },
    "711c4b8dc5034e02bdfb8e862612681b": {
     "model_module": "@jupyter-widgets/controls",
     "model_name": "VBoxModel",
     "state": {
      "_dom_classes": [
       "widget-interact"
      ],
      "_model_module": "@jupyter-widgets/controls",
      "_model_module_version": "1.5.0",
      "_model_name": "VBoxModel",
      "_view_count": null,
      "_view_module": "@jupyter-widgets/controls",
      "_view_module_version": "1.5.0",
      "_view_name": "VBoxView",
      "box_style": "",
      "children": [
       "IPY_MODEL_b124b8ebf77e4eaba729ddf670a35273",
       "IPY_MODEL_66c4d81b40d041f0a8dc07515a51e551",
       "IPY_MODEL_ef4162b0196640e989ff1034ec0e9de3"
      ],
      "layout": "IPY_MODEL_6690247809214b209f881a0bc6951f01"
     }
    },
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         "tags": []
        },
        "output_type": "display_data",
        "text/plain": "<Figure size 720x360 with 1 Axes>"
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