activation_functions.ipynb

activation_functions.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "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": "3.6.4"
    },
    "colab": {
      "name": "activation_functions.ipynb",
      "provenance": [],
      "include_colab_link": true
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/gist/darkdebo/480126b7f5cac2a17ded3f55077a7be7/activation_functions.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2G3kZx728cpD",
        "colab_type": "text"
      },
      "source": [
        "# Activation Functions"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "pPsoUymK8cpI",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "%matplotlib inline  "
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "31kKcgZb8cpO",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "x = np.arange(-5, 5, 0.01)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "50yQpTCb8cpY",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def plot(func, yaxis=(-1.4, 1.4)):\n",
        "    plt.ylim(yaxis)\n",
        "    plt.locator_params(nbins=5)\n",
        "    plt.xticks(fontsize = 14)\n",
        "    plt.yticks(fontsize = 14)\n",
        "    plt.axhline(lw=1, c='black')\n",
        "    plt.axvline(lw=1, c='black')\n",
        "    plt.grid(alpha=0.4, ls='-.')\n",
        "    plt.box(on=None)\n",
        "    plt.plot(x, func(x), c='r', lw=3)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RXPk3vYy8cpf",
        "colab_type": "text"
      },
      "source": [
        "## Binary Step"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "13cvt_Dy8cph",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f(x) = \\left\\{\n",
        "        \\begin{array}{lll}\n",
        "            0 & for & x \\leq 0  \\\\\n",
        "            1 & for & x > 0\n",
        "        \\end{array}\n",
        "    \\right.\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "q-HDqTo48cpi",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f'(x) = \\left\\{\n",
        "        \\begin{array}{lll}\n",
        "            0 & for & x \\neq 0  \\\\\n",
        "            ? & for & x = 0\n",
        "        \\end{array}\n",
        "    \\right.\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Aml1Zw_d8cpj",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "binary_step = np.vectorize(lambda x: 1 if x > 0 else 0, otypes=[np.float])"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "yzLdXuzh8cpp",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "a9e19301-821f-42f1-f515-e926608406f8"
      },
      "source": [
        "plot(binary_step, yaxis=(-0.4, 1.4))"
      ],
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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T5dWLVkee1F9hYuVV3KAHtkpdQGaUlx9t3fSj/goTJa8SB71IbFrRS1Y06EX8aEUv2dKg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\n9x9sAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ewi-xO6v8cpt",
        "colab_type": "text"
      },
      "source": [
        "## Piecewise Linear"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "A_CQh8Va8cpu",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f(x) = \\left\\{\n",
        "        \\begin{array}{lll}\n",
        "            0 & for & x < x_{min} \\\\\n",
        "            mx+b & for & x_{min} \\leq x \\leq x_{max}  \\\\\n",
        "            1 & for & x > x_{max}\n",
        "        \\end{array}\n",
        "    \\right.\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uMWpAdg98cpv",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f'(x) = \\left\\{\n",
        "        \\begin{array}{lll}\n",
        "            0 & for & x < x_{min} \\\\\n",
        "            m & for & x_{min} \\leq x \\leq x_{max}  \\\\\n",
        "            0 & for & x > x_{max}\n",
        "        \\end{array}\n",
        "    \\right.\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "s8ha8J4K8cpw",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "piecewise_linear = np.vectorize(lambda x: 1 if x > 3 else 0 if x < -3 else 1/6*x+1/2, otypes=[np.float])"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ek_buX2b8cpz",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "19b42997-362b-4b71-c223-f106e398b52c"
      },
      "source": [
        "plot(piecewise_linear, yaxis=(-0.4, 1.4))"
      ],
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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lxzd62tjMtgGuBX4NbFb88zozy/0GCasBqwPHAhsD+wE7AFe38Ny5dM14Vefc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\ncbMym0+6T7wAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "MVDJZ90V8cp3",
        "colab_type": "text"
      },
      "source": [
        "## Bipolar"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7pWW16Kh8cp4",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f(x) = \\left\\{\n",
        "        \\begin{array}{lll}\n",
        "            -1 & for & x \\leq 0  \\\\\n",
        "            1 & for & x > 0\n",
        "        \\end{array}\n",
        "    \\right.\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hkuciiwU8cp5",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f'(x) = \\left\\{\n",
        "        \\begin{array}{lll}\n",
        "            0 & for & x \\neq 0  \\\\\n",
        "            ? & for & x = 0\n",
        "        \\end{array}\n",
        "    \\right.\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "J7K-xp_K8cp6",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "bipolar = np.vectorize(lambda x: 1 if x > 0 else -1, otypes=[np.float])"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "GEi38rpF8cp_",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "0b98371f-d736-4472-b020-976c07cd52f9"
      },
      "source": [
        "plot(bipolar)"
      ],
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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ez74/4BN4taSz92NWeSGvXnamLiAzp0i6K3URGQmeV/IBb2bLJS2XpIULF968YsWK+4YO\neVzNR1ktUvOG+Hu9Z/KSM8447fDvfW8SpeIlbssJJ6xZIKmu62Xae7OXpGlJR0jaouZhytcPXf60\nmv49t6qq6bquL5G0/9Axt6p5LfQmSQsknTZ0+QZJ90haXFXVt+u6fpNe/DDrakknSnqyrefkoctH\n/jy1Vqn5V5UPt/UMv3zvkfbrlLaeS2a4jpWS5tR1Pb+t58Shy3/Y1viatp5lM1zHTZIWt7dxqqTj\nhi5fL2ljW+Mjki4aunxXVVU31XV9dntfF0k6euiY+9T8Pzu8refCocufr6pqZV3XU5LuaOs5auiY\neyVtU/P/bKOkqaHLt1dVtaqu6wuqqrqtrefwoWPWSdpU1/VJbT1nD12+taqq1QPXMSXp0N0XVlV1\no2Zgzo3/rdPMPiLpg2MOO985Nz3wPWdJ+q6kE5xzj429kb7MjpZ0woNve9u5r7vuujui3U5hyMvP\nydId652z8Udit7qu3zzb4MGLxcir6wb/KUlfGHPME79mLf04t0HShh/U9Stf96UvfStJDRkiLz8P\nGrMd+ek04J1zmyVtjlwLACCg4I/Bm9mRko7UnsftTjGzBZKecM49E/r2AAAzi/E6+CvUPCnyxfb0\nf7enfz/CbQEAZhF8wDvnrnLO2Qxf14a+rSErI19/acgLsdFjfoLnVdJ70SxKXUBmyAux0WN+gudV\n0oB/IHUBmSEvxEaP+QmeV0kDfvgfUmA08kJs9Jif4Hkx4F+6yAux0WN+GPAAgG4Y8ABQKAY8ABSq\npAH/w9QFZIa8EBs95id4XiUN+CdTF5AZ8kJs9Jif4HmVNOBfk7qAzJAXYqPH/ATPq6QBf3/qAjJD\nXoiNHvMTPK+SBvxMnwiD2ZEXYqPH/ATPq6QBDwAYwIAHgEIx4AGgUAx4AChUSQP+ptQFZIa8EBs9\n5id4XiUN+MWpC8gMeSE2esxP8LxKGvD3pC4gM+SF2OgxP8HzKmnAn5q6gMyQF2Kjx/wEz6ukAX9c\n6gIyQ16IjR7zEzyvkgY8AGAAAx4ACsWAB4BCmXMudQ1BmNly59xnU9eRC/LyQ17+yMxPjLxK2uCX\npy4gM+Tlh7z8kZmf4HmVNOABAAMY8ABQqJIGPI/1+SEvP+Tlj8z8BM+rmCdZAQB7K2mDBwAMYMAD\nQKEY8ABQqGwHvJlNm5kb+rq+w/ddamYPmNlz7X//YBL1pmRmh5rZp83sQTP7lZk9aWb/amavGPN9\n75whY2dmB06q9kkxs3eb2Y/MbLuZrTOzc8Ycf1573HYze9TMrphUrSmZ2ZVm9l0ze9bMfmpmK83s\ntDHfc/wsfXTxpOpOycyumuG+/2TM95xuZt9sf16fMrMPm5n53na2A771n5KOGvj681EHm9kSSTdI\n+qKkM9r/ftnMSv9ggldJerWkD0g6XdLlks6VdF2H792mvTM+yjm3PVKdSZjZZZKukfRRSYsk3Snp\nFjM7dpbjT5C0qj1ukaSPSfq0mV06mYqTmpL0GUlvlHSBpBckfcPMDu3wvRdr7166LVKN+6IfaO/7\nfvpsB5rZyyV9XdJGSb8t6a8kvV/Se71v1TmX5ZekaUn/7Pk9N0j6+tB535B0Xer7kyC/ZZJ2SXr5\niGPeKWlr6lonkMVaSZ8bOu8hSR+b5fiPS3po6Lx/l7Qm9X1JkN18STslXTLimOMlOUlnpa43UUZX\nSbrP4/h3SXpW0kED531I0lNqX/nY9Sv3Df6tZrbZzO43s6vN7OAxxy+RtHrovK+p2UZeal4u6Tk1\nG/ooB5nZ42a2wcy+amaLJlDbxJjZAZLO1Iv7YrVm74vZ+ugsM9s/bIX7vIPVPBLw8w7H3mhmm8zs\nO2b2lsh17WsWmtmP24cBrzezhSOOXSLpW865Xw2c9zU1v4kf73OjOQ/4L0n6I0nnS/pHSZdK+sqY\n7zlSza89gza2579kmNkCNZl9zjn3wohDfyDpTyRVkt4mabuk75jZa+NXOTGHSZojv76YrY/mttf3\nUnKNpO9JWjPimK2S/kbSH6r5zfFWSTeY2eXxy9snrFXz2/DFkv5MTf/cOeI5sNn6a/dlnc31OTg2\nM/uIpA+OOex859y02/td175vZo9KWmtmv+Wc+994Ve47fPIa+J75klaq+XXvA6O+0Tm3RgM/uGZ2\np5of5vdI+st+VaMUZrZC0tmSznbO7ZztOOfcZkn/NHDW3WZ2mJr++0LcKtNzzt0yeNrM/kfSo5Le\nIWlFzNvepwa8pE9p/P/wJ2Y5/241jwW+VtJsA/4nko4YOu+I9vwceeXVDvdV7cnfc55PljrndprZ\n3WoyLsVmNX3j0xez9dEL7fUVz8w+KemtahaIR3tcxVpJfxy2qjw457aa2f2a/edotv7afVln+9SA\nb/+m7/sDcrqaX7WfHnHMGklLJX1i4Lylal4NkR2fvNrnJ26RZJIuds5t9b299mVavynpXt/v3Vc5\n53aY2To1ffDlgYuWavaH/NZIGn557VJJdzvnng9f5b7FzK6RdJma4f5gz6s5Q6N/VovVvsz4dZJu\nn+WQNZI+bmYHDixhSyX9WNJjXjeW+hnmns9Knyjpw5LOUvOkwzJJ69Vs7nMGjrtVA6+EUPOk2QuS\n/rYN+EpJz0tanPo+Rc7r4LZpdm8NRw58HTAir7+T9LuSFqr5gfyPNq83pL5PgfO5TNIOSX8q6WQ1\njytvlXRce/nnJX1+4PgTJP1SzW9QJ7fft0PSpanvywSy+hc1r/C4YKiP5g8c8zFJtw6cfoekt7dZ\n/Yaax+N3SPrr1PdnQpldLem8tm8WS/pqm+Hu/hrO6xA1m/r1kk6T9Ob2+Pd533bqO98zsGMkfVPS\nz9S8EuTh9ofy0KHjHpN07dB5b5H0YNtg6yW9OfX9mUBeU2pepjbT19RseUn6pKTH24w3qXkmf0nq\n+xMpo3e39/85SesknTtw2bSk6aHjz1OzUDwn6UeSrkh9HyaU02x9dNXAMddKemzg9DskPdD+pfis\nmodTL099XyaY2fVqtu8dap77+oqkU2bLqz3vdEl3qHlhw9Nqli2vl0g653g3SQAoVc4vkwQAjMCA\nB4BCMeABoFAMeAAoFAMeAArFgAeAQjHgAaBQDHgAKNT/AxuLQOgjZcrAAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cQRyMGrX8cqD",
        "colab_type": "text"
      },
      "source": [
        "## Sigmoid"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uFh9sUv98cqG",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f(x)={\\frac {1}{1+e^{-x}}}\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mktTmtCs8cqH",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f'(x)=f(x)(1-f(x))\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dWlBWpq48cqI",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def sigmoid(x):\n",
        "    return 1 / (1 + np.exp(-x))"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "gFosHQU38cqM",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "45b8356b-385d-4854-849a-30b67f688aa3"
      },
      "source": [
        "plot(sigmoid, yaxis=(-0.4, 1.4))"
      ],
      "execution_count": 11,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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SrxQb/X1lB1AxypfEpPoqJkq+Umz0e5YdQMUoXxKT6quYKPlKsdGPLzuAilG+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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NYL0zl1n8cqP",
        "colab_type": "text"
      },
      "source": [
        "## Bipolar Sigmoid"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IYdOJcAI8cqQ",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f(x)={\\frac {1-e^{-x}}{1+e^{-x}}}\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "VApYvct38cqR",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f'(x)={\\frac {2e^x}{(e^x+1)^2}}\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "xwCxpTjr8cqS",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def bipolar_sigmoid(x):\n",
        "    return (1 - np.exp(-x)) / (1 + np.exp(-x))"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "c_Snp5ao8cqW",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "d66a594c-990f-4197-e320-a658afa7a87b"
      },
      "source": [
        "plot(bipolar_sigmoid)"
      ],
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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EpVTgdayvGOWrGOWrOOWsmNLzlcxJVhER6SulEbyIiOSowIuIJEoFXkQkUbUt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\nEZFEqcCLiCRKBV5EJFEq8CIiifr/KTCJZmp6xPAAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3DzFgK4H8cqZ",
        "colab_type": "text"
      },
      "source": [
        "## Hyperbolic Tangent, TanH"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uGdkI01L8cqa",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f(x)={\\frac {2}{1+e^{-2x}}}-1\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tWxmqyKF8cqb",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f'(x)=1-f(x)^2\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Egd1F9j38cqc",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def tanh(x):\n",
        "    return 2 / (1 + np.exp(-2 * x)) -1"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "uRtavjZm8cqg",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "e9157c37-0dc5-4324-fee0-3171c0f01d86"
      },
      "source": [
        "plot(tanh)"
      ],
      "execution_count": 15,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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EvDaAda83s6VmdoeZfaTkuFrNeDP7Q7Yb8GozG9/HulOB/3XOvZ0b+xXpX+I7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\nFxGJlBq8iEik1OBFRCL1f1QjhM2vg4loAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yZ21GKfF8cqq",
        "colab_type": "text"
      },
      "source": [
        "## Arctangent, ArcTan"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "veMdHk7R8cqu",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f(x)=tan^{-1}(x)\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "OwxPPyxK8cqv",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f'(x)={\\frac {1}{1+x^2}}\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "57Fgr1Gj8cqx",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def arctan(x):\n",
        "    return np.arctan(x)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "E184h5n68cq1",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "3a3031dc-1a23-43e8-e08a-6c55b9761b15"
      },
      "source": [
        "plot(arctan)"
      ],
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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CBV6kCLNXAgfllujmqnSslAr8M7EDqBnla2Q+QO+QfLfgfnPMYDqc2lgxpedL3QWLDIOZ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\nR8UAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wpOYHtTc8cq3",
        "colab_type": "text"
      },
      "source": [
        "## Rectified Linear Units, ReLU"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IzgAq6yJ8cq4",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f(x) = \\left\\{\n",
        "        \\begin{array}{lll}\n",
        "            0 & for & x \\leq 0  \\\\\n",
        "            x & for & x > 0\n",
        "        \\end{array}\n",
        "    \\right.\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8kM27cYW8cq6",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f'(x) = \\left\\{\n",
        "        \\begin{array}{lll}\n",
        "            0 & for & x \\leq 0  \\\\\n",
        "            1 & for & x > 0\n",
        "        \\end{array}\n",
        "    \\right.\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "1cMIFsCW8cq7",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "relu = np.vectorize(lambda x: x if x > 0 else 0, otypes=[np.float])"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "YbqmylZj8cq-",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "193089a8-d4c8-4be8-ebb9-e07357c33386"
      },
      "source": [
        "plot(relu, yaxis=(-0.4, 1.4))"
      ],
      "execution_count": 19,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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VMIV9e+AjthZFsWbk8Mp2u70IOKxrTAeYUWUxjTJb5t9447Vvf+KJ3wJwZkvv/MY3VlT1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\ngg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7MclVA1s8crB",
        "colab_type": "text"
      },
      "source": [
        "## Leaky Rectified Linear Units, Leaky ReLU"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3lqFTX8u8crC",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f(x) = \\left\\{\n",
        "        \\begin{array}{lll}\n",
        "            ax & for & x \\leq 0  \\\\\n",
        "            x & for & x > 0\n",
        "        \\end{array}\n",
        "    \\right.\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "lPUROHm88crD",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f'(x) = \\left\\{\n",
        "        \\begin{array}{lll}\n",
        "            a & for & x \\leq 0  \\\\\n",
        "            1 & for & x > 0\n",
        "        \\end{array}\n",
        "    \\right.\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "65CQibYz8crE",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "leaky_relu = np.vectorize(lambda x: max(0.1 * x, x), otypes=[np.float])"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "eazlFtim8crH",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "7cde7da9-f162-4e96-bc0a-a5a340ad5c5d"
      },
      "source": [
        "plot(leaky_relu)"
      ],
      "execution_count": 21,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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N7BbgB8DHqQ7d/J9I65PJzOxgOnvoC+ncjGMsG+g09GuBh3BOY3ZIcaI0eOfcFWb2BuBP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\n/wEa9UUwPFrPpAAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "spayCR4e8crL",
        "colab_type": "text"
      },
      "source": [
        "Parametric ReLU is similar to Leaky ReLU but coefficient of leakage is learned as parameter of neural network"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "5abzYQSh8crN",
        "colab_type": "text"
      },
      "source": [
        "## Exponential Linear Units, ELU"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "T-iGi3Ub8crP",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f(x) = \\left\\{\n",
        "        \\begin{array}{lll}\n",
        "            a(e^x-1) & for & x \\leq 0  \\\\\n",
        "            x & for & x > 0\n",
        "        \\end{array}\n",
        "    \\right.\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JJnloxNZ8crR",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f'(x) = \\left\\{\n",
        "        \\begin{array}{lll}\n",
        "            f(x)+a & for & x \\leq 0  \\\\\n",
        "            1 & for & x > 0\n",
        "        \\end{array}\n",
        "    \\right.\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "k41XS9NE8crS",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "elu = np.vectorize(lambda x: x if x > 0 else 0.5 * (np.exp(x) - 1), otypes=[np.float])"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "_ZIOVPcI8crX",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "06175093-d983-4e68-b190-5599262154e0"
      },
      "source": [
        "plot(elu)"
      ],
      "execution_count": 23,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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zTidWJUaNNHgA59zlwOVNfb5IrcxOBb5D7zbxIDAX57aEC0pk96R0L5p1oQOIjPLVw2wm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\nRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Ys9pWJC58cre",
        "colab_type": "text"
      },
      "source": [
        "## SoftPlus"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_GoTwVr08crf",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f(x)=ln(1+e^x)\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yDdqOsJk8crg",
        "colab_type": "text"
      },
      "source": [
        "$$\n",
        "f'(x)={\\frac {1}{1+e^{-x}}}\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ZHUyXDrl8crh",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def softplus(x):\n",
        "    return np.log(1+np.exp(x))"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "gYVs8DlB8crj",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "170fbaa7-2dd9-4278-fae9-314b40e94dd9"
      },
      "source": [
        "plot(softplus, yaxis=(-0.4, 1.4))"
      ],
      "execution_count": 25,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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j2lV7O5nZ2Wb2kpmtMLPHzewjAyx/RL7cCjN70czOaletMZnZBWb2UzNbYma/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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    }
  ]
}