1_1_BackgroundMathematics.ipynb

{
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  "metadata": {
    "colab": {
      "provenance": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "jbang-ijava",
      "display_name": "java (IJava/j!)"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/jamalromero/colab/blob/main/1_1_BackgroundMathematics.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# **Understanding Deep Learning** [book](https://udlbook.github.io/udlbook/)\n",
        "## **Notebook 1.1 -- Background Mathematics**"
      ],
      "metadata": {
        "id": "xIr_Md1WRvoi"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Keep Python runtime and install java"
      ],
      "metadata": {
        "id": "-PY9c_q42J1S"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from google.colab import drive\n",
        "drive.mount('/content/drive')"
      ],
      "metadata": {
        "id": "pjUM1cc8pP04"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "tGjdJncAIbnT"
      },
      "outputs": [],
      "source": [
        "!apt install openjdk-17-jdk"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Install Juyter Java kernel"
      ],
      "metadata": {
        "id": "VTvVS96Q2Tzd"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "!pip install jbang\n",
        "import jbang\n",
        "jbang.exec(\"trust add https://github.com/jupyter-java\")\n",
        "jbang.exec(\"install-kernel@jupyter-java\")"
      ],
      "metadata": {
        "id": "advvIMn1Nn0e"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Change Runtime to Java kernel and test Java runtime."
      ],
      "metadata": {
        "id": "ro6LZ1iT2cig"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "System.getProperty(\"java.version\")"
      ],
      "metadata": {
        "id": "cwZaX7xHNcKk"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Load external script to include maven dependency, import packages and intial startup script. Example:\n",
        "\n",
        "```\n",
        "%%loadFromPOM\n",
        "<dependency>\n",
        "    <groupId>tech.tablesaw</groupId>\n",
        "    <artifactId>tablesaw-core</artifactId>\n",
        "    <version>0.43.1</version>\n",
        "</dependency>\n",
        "\n",
        "import java.util.stream.*;\n",
        "\n",
        "System.getProperty(\"java.version\");\n",
        "\n",
        "```\n",
        "\n"
      ],
      "metadata": {
        "id": "z6jlXMb82p9i"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "%load /content/drive/MyDrive/colab/init_script.ipynb"
      ],
      "metadata": {
        "id": "HYw6JGoL4A29"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "WV2Dl6owme2d"
      },
      "source": [
        "##Linear functions\n",
        "We will be using the term *linear equation* to mean a weighted sum of inputs plus an offset. If there is just one input $x$, then this is a straight line:\n",
        "\n",
        "\\begin{equation}y=\\beta+\\omega x,\\end{equation}\n",
        "\n",
        "where $\\beta$ is the y-intercept of the linear and $\\omega$ is the slope of the line. When there are two inputs $x_{1}$ and $x_{2}$, then this becomes:\n",
        "\n",
        "\\begin{equation}y=\\beta+\\omega_1 x_1 + \\omega_2 x_2.\\end{equation}\n",
        "\n",
        "Any other functions are by definition **non-linear**.\n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 1D Linear function\n",
        "\n",
        "* Define an array of x values from 0 to 10 with increments of 0.01\n",
        "* Compute y using the function: \\begin{equation}y=\\beta+\\omega x\\end{equation}\n",
        "* Plot the function\n",
        "* Experiment with changing the values of $\\beta$ and  $\\omega$\n",
        "* Make a line that crosses the y-axis at 10 and the x-axis at 5"
      ],
      "metadata": {
        "id": "sVjxquFlQ01E"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "NDManager manager = NDManager.newBaseManager()"
      ],
      "metadata": {
        "id": "SuUbShGttDiJ"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "var beta = 10f;\n",
        "var omega = -2f;\n",
        "var x = manager.arange(0f, 10f, 0.01f, DataType.FLOAT64);\n",
        "var y = x.mul(omega).add(beta);\n",
        "var data = toDoubleArray(x.stack(y, 1));\n",
        "var canvas = LinePlot.of(data, Color.BLUE).canvas();\n",
        "canvas.setBound(new double[]{0, 0}, new double[]{8, 13});\n",
        "canvas.toBufferedImage(600, 400);"
      ],
      "metadata": {
        "id": "CRs6UXYgtMq_",
        "outputId": "9ee57505-b5a1-4b46-b7dd-18b0bbcf3ef4",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 417
        }
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
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",
            "text/plain": [
              "BufferedImage@64464664: type = 2 DirectColorModel: rmask=ff0000 gmask=ff00 bmask=ff amask=ff000000 IntegerInterleavedRaster: width = 600 height = 400 #Bands = 4 xOff = 0 yOff = 0 dataOffset[0] 0"
            ]
          },
          "metadata": {},
          "execution_count": 148
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "AedfvD9dxShZ"
      },
      "source": [
        "## 2D linear function\n",
        "\n",
        "* Define 2 arrays $x_1$ and $x_2$ with values from 0 to 10 with increments of 0.1\n",
        "* Define a linear function with two inputs, $x_1$ and $x_2$ \\begin{equation}y=\\beta+\\omega_1 x_1 + \\omega_2 x_2\\end{equation}\n",
        "* Plot the 2D function\n",
        "* Predict what this plot will look like if you set $\\omega_1 = 0$\n",
        "* Predict what this plot will look like if you set $\\omega_2 = 0$\n",
        "* Predict what this plot will look like if you set $\\beta = -5$"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "var beta = -5f;\n",
        "var omega1 = 1f;\n",
        "var omega2 = -0.5f;\n",
        "var x1 = manager.arange(0f, 10f, 0.1f, DataType.FLOAT64);\n",
        "var x2 = manager.arange(0f, 10f, 0.1f, DataType.FLOAT64);\n",
        "var gridMap = meshgrid(x1, x2);\n",
        "x1_mesh = gridMap.get(\"x1\");\n",
        "x2_mesh = gridMap.get(\"x2\");\n",
        "var z = x1_mesh.mul(omega1).add(x2_mesh.mul(omega2)).add(beta);\n",
        "var canvas = Heatmap.of(x1.toDoubleArray(), x2.toDoubleArray(), toDoubleArray(z), 256).canvas();\n",
        "canvas.toBufferedImage(600, 400);"
      ],
      "metadata": {
        "id": "iZbSpOrcyync",
        "outputId": "59c372fe-5521-4391-dd2d-12b40ba1b6ee",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 417
        }
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": 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",
            "text/plain": [
              "BufferedImage@79febdb1: type = 2 DirectColorModel: rmask=ff0000 gmask=ff00 bmask=ff amask=ff000000 IntegerInterleavedRaster: width = 600 height = 400 #Bands = 4 xOff = 0 yOff = 0 dataOffset[0] 0"
            ]
          },
          "metadata": {},
          "execution_count": 182
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "i8tLwpls476R"
      },
      "source": [
        "Often we will want to compute many linear functions at the same time.  For example, we might have three inputs, $x_1$, $x_2$, and $x_3$ and want to compute two linear functions giving $y_1$ and $y_2$. Of course, we could do this by just running each equation separately,<br><br>\n",
        "\n",
        "\\begin{align}y_1 &=& \\beta_1 + \\omega_{11} x_1 + \\omega_{12} x_2 + \\omega_{13} x_3\\\\\n",
        "y_2 &=& \\beta_2 + \\omega_{21} x_1 + \\omega_{22} x_2 + \\omega_{23} x_3\n",
        "\\end{align}\n",
        "\n",
        "However, we can write it more compactly with vectors and matrices:\n",
        "\n",
        "\\begin{equation}\n",
        "\\begin{bmatrix} y_1\\\\ y_2 \\end{bmatrix} = \\begin{bmatrix}\\beta_{1}\\\\\\beta_{2}\\end{bmatrix}+ \\begin{bmatrix}\\omega_{11}&\\omega_{12}&\\omega_{13}\\\\\\omega_{21}&\\omega_{22}&\\omega_{23}\\end{bmatrix}\\begin{bmatrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{bmatrix}\n",
        "\\end{equation}\n",
        "or\n",
        "\n",
        "\\begin{equation}\n",
        "\\mathbf{y} = \\boldsymbol\\beta +\\boldsymbol\\Omega\\mathbf{x}\n",
        "\\end{equation}\n",
        "\n",
        "for short.  Here, lowercase bold symbols are used for vectors.  Upper case bold symbols are used for matrices.\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 3D Linear function"
      ],
      "metadata": {
        "id": "6C_cYzWoiXBe"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "var beta_vec = manager.create(new float[]{0.5f, 0.2f});\n",
        "var omega_mat = manager.create(new float[][]{{-1f, 0.4f, -0.3f}, {0.1f, 0.1f, 1.2f}});\n",
        "var x_vec = manager.create(new float[]{4f, -1f, 2f});\n",
        "var y_vec = beta_vec.add(omega_mat.matMul(x_vec));\n",
        "y_vec"
      ],
      "metadata": {
        "id": "KeVWAr8g7KWy",
        "outputId": "324a810c-f762-4b51-be51-40b2961530af",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "ND: (2) cpu() float32\n",
              "[-4.5,  2.9]\n"
            ]
          },
          "metadata": {},
          "execution_count": 267
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3LGRoTMLU8ZU"
      },
      "source": [
        "## Questions\n",
        "\n",
        "1.  A single linear equation with three inputs (i.e. **linear_function_3D()**) associates a value y with each point in a 3D space ($x_1$,$x_2$,$x_3$).  Is it possible to visualize this?   What value is at position (0,0,0)?\n",
        "\n",
        "2.  Write code to compute three linear equations with two inputs ($x_1$, $x_2$) using both the individual equations and the matrix form (you can make up any values for the inputs $\\beta_{i}$ and the slopes $\\omega_{ij}$."
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "x_vec = manager.zeros(new Shape(3));\n",
        "y_vec = beta_vec.add(omega_mat.matMul(x_vec));\n",
        "y_vec"
      ],
      "metadata": {
        "id": "Sf-8EoVA-Del",
        "outputId": "1f1335ef-658c-49f6-9921-f1ef685749d4",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "ND: (2) cpu() float32\n",
              "[0.5, 0.2]\n"
            ]
          },
          "metadata": {},
          "execution_count": 269
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "var beta_vec = manager.create(new float[]{0.5f, 0.2f, -1f});\n",
        "var omega_mat = manager.create(new float[][]{{-1f, 0.4f}, {0.1f, 0.1f}, {-0.1f, 1f}});\n",
        "var x_vec = manager.create(new float[]{4f, -1f});\n",
        "var y_vec = beta_vec.add(omega_mat.matMul(x_vec));\n",
        "y_vec"
      ],
      "metadata": {
        "id": "0u9PQeNP_Z6d",
        "outputId": "4adae2ae-46c5-40c4-9132-95477f1c4f8e",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "ND: (3) cpu() float32\n",
              "[-3.9,  0.5, -2.4]\n"
            ]
          },
          "metadata": {},
          "execution_count": 272
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7Y5zdKtKZAB2"
      },
      "source": [
        "## Special functions\n",
        "\n",
        "Throughout the book, we'll be using some special functions (see Appendix B.1.3).  The most important of these are the logarithm and exponential functions.  Let's investigate their properties.\n",
        "\n",
        "We'll start with the exponential function $y=\\exp[x]=e^x$ which maps the real line $[-\\infty,+\\infty]$ to non-negative numbers $[0,+\\infty]$."
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "var x = manager.arange(-5f, 5f, 0.01f, DataType.FLOAT64);\n",
        "var y = x.exp();\n",
        "LinePlot.of(toDoubleArray(x.stack(y, 1))).canvas().toBufferedImage(600, 400);"
      ],
      "metadata": {
        "id": "4nJZqhKEACE6",
        "outputId": "a58018d6-4348-4e64-85b2-615034eaa465",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 417
        }
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": 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",
            "text/plain": [
              "BufferedImage@58a67e2c: type = 2 DirectColorModel: rmask=ff0000 gmask=ff00 bmask=ff amask=ff000000 IntegerInterleavedRaster: width = 600 height = 400 #Bands = 4 xOff = 0 yOff = 0 dataOffset[0] 0"
            ]
          },
          "metadata": {},
          "execution_count": 289
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XyrT8257IWCu"
      },
      "source": [
        "## Questions\n",
        "\n",
        "1. What is $e^0$?  \n",
        "2. What is $e^1$?\n",
        "3. What is $e^{-\\infty}$?\n",
        "4. What is $e^{+\\infty}$?\n",
        "5. A function is convex if we can draw a straight line between any two points on the function, and the line lies above the function everywhere between these two points. Similarly, a function is concave if a straight line between any two points lies below the function everywhere between these two points.  Is the exponential function convex or concave or neither?\n"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "System.out.print(manager.create(new float[]{0f}).exp());\n",
        "System.out.print(manager.create(new float[]{1f}).exp());\n",
        "System.out.print(manager.create(new float[]{Float.NEGATIVE_INFINITY}).exp());\n",
        "System.out.print(manager.create(new float[]{Float.POSITIVE_INFINITY}).exp());"
      ],
      "metadata": {
        "id": "BbcyIEhuEvQE",
        "outputId": "bf07521e-4d72-46c9-9f9f-00e6862b5a6e",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "ND: (1) cpu() float32\n",
            "[1.]\n",
            "ND: (1) cpu() float32\n",
            "[2.7183]\n",
            "ND: (1) cpu() float32\n",
            "[0.]\n",
            "ND: (1) cpu() float32\n",
            "[inf]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Yes, the function $e^x$ is convex. Let's verify this using the second derivative test:\n",
        "\n",
        "### 1. **First Derivative:**\n",
        "The first derivative of $f(x) = e^x$ is:\n",
        "\n",
        "$f'(x) = \\frac{d}{dx} e^x = e^x$\n",
        "\n",
        "### 2. **Second Derivative:**\n",
        "The second derivative of $f(x) = e^x$ is:\n",
        "\n",
        "$f''(x) = \\frac{d}{dx} e^x = e^x$\n",
        "\n",
        "### 3. **Convexity Condition:**\n",
        "A function $f(x)$ is convex if its second derivative $f''(x)$ is non-negative for all $x$. Since $e^x > 0$ for all $x$, the second derivative $f''(x) = e^x$ is always positive.\n",
        "\n",
        "### Conclusion:\n",
        "Since the second derivative of $e^x$ is always positive, $e^x$ is a strictly convex function."
      ],
      "metadata": {
        "id": "XBo2KzHbvJ6h"
      }
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yYWrL5AXIWCv"
      },
      "source": [
        "## Questions\n",
        "\n",
        "1. What is $log[0]$?  \n",
        "2. What is $log[1]$?\n",
        "3. What is $log[e]$?\n",
        "4. What is $log[e^3]$?\n",
        "5. What is $e^{log[4]}$?\n",
        "6. What is $log[-1]$?\n",
        "7. Is the logarithm function concave or convex?\n"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "System.out.print(manager.create(new float[]{0f}).log());\n",
        "System.out.print(manager.create(new float[]{1f}).log());\n",
        "System.out.print(manager.create(new double[]{Math.E}).log());\n",
        "System.out.print(manager.create(new float[]{3f}).exp().log());\n",
        "System.out.print(manager.create(new float[]{4f}).log().exp());\n",
        "System.out.print(manager.create(new float[]{-1f}).log());"
      ],
      "metadata": {
        "id": "z7B1lIweF9KS",
        "outputId": "162b20ca-c30e-4d0e-e7d8-f167b0201527",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "ND: (1) cpu() float32\n",
            "[-inf]\n",
            "ND: (1) cpu() float32\n",
            "[0.]\n",
            "ND: (1) cpu() float64\n",
            "[1.]\n",
            "ND: (1) cpu() float32\n",
            "[3.]\n",
            "ND: (1) cpu() float32\n",
            "[4.]\n",
            "ND: (1) cpu() float32\n",
            "[nan]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "var x = manager.arange(0.01f, 5f, 0.01f, DataType.FLOAT64);\n",
        "var y = x.log();\n",
        "LinePlot.of(toDoubleArray(x.stack(y, 1))).canvas().toBufferedImage(600, 400);"
      ],
      "metadata": {
        "id": "faP4S8RjGfNM",
        "outputId": "44ba9ccb-2edd-4198-ba88-f6b588dceab6",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 417
        }
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": 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",
            "text/plain": [
              "BufferedImage@7adaf1d: type = 2 DirectColorModel: rmask=ff0000 gmask=ff00 bmask=ff amask=ff000000 IntegerInterleavedRaster: width = 600 height = 400 #Bands = 4 xOff = 0 yOff = 0 dataOffset[0] 0"
            ]
          },
          "metadata": {},
          "execution_count": 323
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Yes, the function $log(x)$ is **concave** and not convex.\n",
        "\n",
        "To determine whether a function is convex or concave, you can examine its second derivative:\n",
        "\n",
        "1. The first derivative of $log(x)$ with respect to $x$ is:\n",
        "   $\\frac{d}{dx} log(x) = \\frac{1}{x}$\n",
        "\n",
        "2. The second derivative is: $\\frac{d^2}{dx^2} \\log(x) = -\\frac{1}{x^2}$\n",
        "\n",
        "Since the second derivative, $-\\frac{1}{x^2}$, is always negative for $x > 0$, $log(x)$ is a concave function on its domain $x > 0$.\n",
        "\n",
        "A function is convex if its second derivative is non-negative $\\geq 0$ for all $x$ in its domain. Since $log(x)$ has a negative second derivative, it is concave, not convex."
      ],
      "metadata": {
        "id": "VUGGaCGiJwq4"
      }
    }
  ]
}