Basics_of_Linear_Algebra_for_NLP.ipynb

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  "metadata": {
    "colab": {
      "name": "Basics_of_Linear_Algebra_for_ML.ipynb",
      "provenance": [],
      "collapsed_sections": [],
      "authorship_tag": "ABX9TyMunz4OKaJ17UIzFA65onoU",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/Taaniya/linear-algebra-for-ml/blob/main/Basics_of_Linear_Algebra_for_ML.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IlWGBGiOL2mV"
      },
      "source": [
        "## From Linear Algebra to Text representation in NLP\n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "1XyMykY_FO_j"
      },
      "source": [
        "import numpy as np\n",
        "from matplotlib import pyplot as plt\n",
        "plt.rcParams[\"figure.figsize\"] = [7.00, 3.50]\n",
        "plt.rcParams[\"figure.autolayout\"] = True"
      ],
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "vDgV_FnccVeF"
      },
      "source": [
        "#### Unit vectors of our coordinate system\n",
        "$\\hat{i} $ -> Denotes a unit vector (vector of length 1 unit) pointing in the x-direction \n",
        "\n",
        "$\\hat{j} $ -> Denotes a unit vector in the y-direction \n",
        "\n",
        "We can represent any vector say $ a = (a_1, a_2) $ in terms of these standard unit vectors\n",
        "\n",
        "$ a = (a_1, a_2) = a_1 \\hat i + a_2 \\hat j$\n",
        "\n",
        "$a_1$ and $ a_2 $ are the x,y components of the vector\n",
        "\n",
        "Together, they are called the basis of our coordinate vector space.\n",
        "\n",
        "We'll come to the term **basis** more in the subsequent parts of this notebook. \n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 261
        },
        "id": "TTnEtay-WLjE",
        "outputId": "f4f4c664-fb0c-42d9-ba35-715832af7f2d"
      },
      "source": [
        "data = np.array([[1, 0], [0, 1]])              # List of x,y coordinates for each vector head\n",
        "origin = np.array([[0, 0], [0, 0]])            # List of x,y coordinates for each vector origin\n",
        "plt.ylim(-2,3)\n",
        "plt.xlim(-2,3)\n",
        "plt.quiver(*origin, data[:, 0], data[:, 1], color=['black', 'cyan'], angles='xy', scale_units='xy', scale=1.)\n",
        "plt.grid()\n",
        "plt.xlabel(\"x-axis\")\n",
        "plt.ylabel(\"y-axis\")\n",
        "plt.show()"
      ],
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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\n",
            "text/plain": [
              "<Figure size 504x252 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3jdkQpn5mv-u"
      },
      "source": [
        "* Suppose we have a vector $ 3\\hat{i}$ + 5$\\hat{j}$. \n",
        "* This vector has x,y coordinates : 3 & 5 respectively.\n",
        "* These coordinates are the scalars that flip and scale the unit vectors by 3 & 5 units in the x & y directions respectively.\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 261
        },
        "id": "Rarc_dTlsvEq",
        "outputId": "9da27c16-c3e2-43bb-f930-06669ab4b89b"
      },
      "source": [
        "data = np.array([[3, 5]])              # List of x,y coordinates for each vector head\n",
        "origin = np.array([[0, 0]])            # List of x,y coordinates for each vector origin\n",
        "plt.ylim(-2,6)\n",
        "plt.xlim(-2,6)\n",
        "plt.quiver(origin[:,0], origin[:,1], data[:, 0], data[:, 1], color=['black'], angles='xy', scale_units='xy', scale=1.)\n",
        "plt.grid()\n",
        "plt.xlabel(\"x-axis\")\n",
        "plt.ylabel(\"y-axis\")\n",
        "plt.show()"
      ],
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 504x252 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Ixw9m5aTduU0"
      },
      "source": [
        "\n",
        "### Linear Combination of 2 vectors\n",
        "\n",
        "If $ \\hat{u}$ & $\\hat{v}$ are two vectors in a 2 dimensional space,then their linear combination resulting into a vector $\\hat{l} $ is represented by -\n",
        "\n",
        "$\\hat{l} = x_1 u + x_2 v$ \n",
        "\n",
        "* The numbers $x_1$, $x_2$ are the components of a vector x\n",
        "* This is essentially a scaling and addition operation by x on the given vectors. \n",
        "\n",
        "The above expression of linear combination is equivalent to the following linear system -\n",
        "\n",
        "$Bx = \\hat{l}$\n",
        "\n",
        "Where B denotes a matrix whose columns are $u$ & $v$.\n",
        "\n",
        "Let's understand this by an example below with vectors u & v in a 2 dimensional space -"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 314
        },
        "id": "AvOzNyjNWOpT",
        "outputId": "da798c55-6b9e-4f45-ca2d-0be43d87bcfb"
      },
      "source": [
        "# Vectors u & v\n",
        "u_vec = np.array([1, -1, 0])              # The vectors are 3D, we'll only use 2 dimensions\n",
        "v_vec = np.array([0, 1, -1])\n",
        "print(f\" shape of vector u - {u_vec.shape}\")\n",
        "\n",
        "# Vector x\n",
        "x_vec = np.array([1.5, 2])\n",
        "\n",
        "# Let's plot to see how they look in the vector space\n",
        "data = np.vstack((u_vec, v_vec))[:,:2]              # fetch coords from first 2 dimensions\n",
        "origin = np.array([[0, 0, 0], [0, 0, 0]])[:,:2]\n",
        "plt.ylim(-2,3)\n",
        "plt.xlim(-2,3)\n",
        "QV = plt.quiver(origin[:,0],origin[:,1], data[:, 0], data[:, 1], color=['black', 'green'], angles='xy', scale_units='xy', scale=1.)\n",
        "plt.grid()\n",
        "plt.xlabel(\"x-axis\")\n",
        "plt.ylabel(\"y-axis\")\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            " shape of vector u - (3,)\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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Q2HMoKCAD5efna8GCBfrNb36jQw45RJJUVVWl8ePHa+DAgSotLQ2cEPjqKCggQ5mZRo8erfLycl100UX160tLSzVgwABdeeWV+uijjwImBL4aCgrIcAcffLDuu+8+LVq0SN27d5ck1dbWasqUKerevbtmz57NU3yRkSgoYC8xePBgrVixQrfccovatWsnSXrrrbc0cuRInXXWWXrzzTc/9zVbt25t6ZhAYhQUsBdp27atfvSjH2nNmjU6/fTT69fPmTNHPXr00F133aWampr69UVFRdq4cWOIqECzKChgL9S1a1c988wzeuSRR9SxY0dJ0scff6yJEyfqxBNP1NKlSyVJK1eu1LBhw/Tee++FjAs0ioIC9lJmplGjRqmsrEyXXnpp/foVK1bopJNO0tixY/XBBx9o/fr1GjFihD755JOAaYHPo6CAvdyBBx6oadOm6cUXX1SvXr0kSe6uadOm1e/z0ksvafTo0byHClGhoIAsMWjQIC1fvly333672rdv/7ntjz/+uCZOnBggGdA4CgrIIjk5OTrmmGPqHzO/q8mTJ+vnP/95C6cCGkdBAVmipqZGl112mYqKinb78MMJEybosccea8FkQOOCFJSZFZnZWjOrNbP+ITIA2SYnJ0fTpk1TWVmZpkyZojPPPFP77rvv5/Zzd40ePVqLFy8OkBL4u1BHUGskjZS0KNDrA1nJzNStWzddccUVmjt3rt5//32VlJRo0qRJOvHEE2VmkqTq6mqNGDFC5eXlgRMjmwUpKHcvc/eKEK8N4O/atm2rU089VTfffLOWLl2qLVu2aObMmbr44ovVoUMHFRYW6p133gkdE1kqJ3QAAPE4+OCDVVRUpKKiIrm7Xn31VW3YsCF0LGSptBWUmc2X1KmRTZPc/Ykv8H2KJRVLdU8WLSkp2TMB92JVVVWMU0KMVTKMUzKM055lIe9ybGYlkia6+7Ik++fn53tFBWcGm1NSUqIhQ4aEjpERGKtkGKdkGKdkzKzU3ZudIMc0cwBAlEJNMz/bzDZJOlnSU2b2bIgcAIB4BZkk4e6zJc0O8doAgMzAKT4AQJQoKABAlCgoAECUKCgAQJQoKABAlCgoAECUKCgAQJQoKABAlCgoAECUKCgAQJQoKABAlCgoAECUKCgAQJQoKABAlCgoAECUKCgAQJQoKABAlCgoAECUKCgAQJQoKABAlCgoAECUKCgAQJQoKABAlCgoAECUKCgAQJQoKABAlCgoAECUKCgAQJQoKABAlCgoAECUKCgAQJQoKABAlCgoAECUKCgAQJQoKABAlCgoAECUKCgAQJQoKABAlIIUlJndYWblZrbKzGab2QEhcgAA4hXqCOo5Sb3cvY+k9ZKuCZQDABCpIAXl7vPcvSa1+EdJXULkAADEKyd0AEljJM1oaqOZFUsqTi1Wm9maFkmV2Q6RtDV0iAzBWCXDOCXDOCWTn2Qnc/e0vLqZzZfUqZFNk9z9idQ+kyT1lzTSEwQxs2Xu3n/PJt37ME7JMVbJME7JME7JJB2ntB1Bufu/7G67mV0o6TuS/jlJOQEAskuQU3xmVijpKkmnuvu2EBkAAHELNYvvF5I6SHrOzFaY2bSEX3dvGjPtTRin5BirZBinZBinZBKNU9quQQEA8FVwJwkAQJQoKABAlDKuoLhNUjJmVmRma82s1syY9roLMys0swoze83Mfhg6T6zM7AEze5f3H+6emR1uZgvNbF3q39340JliZGbtzGypma1MjdNPdrd/xhWUuE1SUmskjZS0KHSQ2JhZa0lTJQ2T1EPSKDPrETZVtB6UVBg6RAaokTTB3XtIOknSf/Az1ahqSae5+3GSjpdUaGYnNbVzxhUUt0lKxt3L3L0idI5IDZD0mrtvdPdPJT0q6azAmaLk7oskvR86R+zc/W13X576/CNJZZI6h00VH69TlVpsk/pocqZexhXULsZI+r/QIZBxOkv6S4PlTeI/E+whZnaUpL6SloRNEicza21mKyS9K+k5d29ynGK4F9/nfIHbJNVImt6S2WKSZJwAtBwzy5X0uKQr3f3D0Hli5O47JR2fmj8w28x6uXuj1zijLChuk5RMc+OEJr0l6fAGy11S64AvzczaqK6cprv7rNB5YufulWa2UHXXOBstqIw7xdfgNkkjuE0SvqRXJH3TzI42s7aSzpP0ZOBMyGBmZpLul1Tm7j8LnSdWZpb32cxrM2svaaik8qb2z7iC0pe/TVJWMbOzzWyTpJMlPWVmz4bOFIvUJJvLJT2ruovZM919bdhUcTKz30l6WVK+mW0ys4tCZ4rUKZJGSzot9f/SCjM7I3SoCH1d0kIzW6W6XxSfc/e5Te3MrY4AAFHKxCMoAEAWoKAAAFGioAAAUaKgAABRoqAAAFGioIAMYWaXmdn3Q+cAWgrTzAEAUeIICkgDMzsx9cyydma2b+rZN7122We4mS0xsz+Z2Xwz65haP8XMrkt9/m0zW2RmrczsBjObmFp/RerZQ6vM7NGW/xsC6ccRFJAmZnazpHaS2kva5O7/vcv2AyVVurub2cWSurv7BDPbR3Xvsr9c0jRJZ7j7BjO7QVKVu99pZpslHe3u1WZ2gLtXtuTfDWgJUd4sFthL3Ki6otku6YpGtneRNMPMvi6praTXJcndt5nZJap72OR/uvuGRr52laTpZvZ7Sb9PR3ggNE7xAelzsKRc1d07sp2Z3fLZfdpS2/9H0i/cvbekS1V3tPWZ3pLek3RYE9/7TNU9FfgESa+YGb9sYq9DQQHpc4+kH6vumWW3u/skdz/e3Y9Pbd9ff3/MxwWffZGZHSlpguoeejfMzAY2/KZm1krS4e6+UNLVqe+Tm9a/CRAAv3UBaZCaDr7D3R8xs9aSXjKz09z9+Qa73SDpMTP7QNLzko5u8NiGie6+OXX38AfN7MQGX9da0sNmtr8kk3Q316CwN2KSBAAgSpziAwBEiYICAESJggIARImCAgBEiYICAESJggIARImCAgBE6f8Bbzt5+chITQAAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 314
        },
        "id": "YkGI3FD20FF5",
        "outputId": "1a066a27-e9c5-4263-842a-523735517cc8"
      },
      "source": [
        "# Finding their linear combination \n",
        "uv_matrix = np.vstack((u_vec, v_vec)).transpose()          # Collecting them into a matrix \n",
        "print(f\" shape of matrix containing u & v - {uv_matrix.shape}\")\n",
        "\n",
        "l_vec = np.dot(uv_matrix, x_vec)\n",
        "\n",
        "data = np.vstack((u_vec, v_vec, l_vec))[:,:2]              # fetch coords from first 2 dimensions\n",
        "origin = np.array([[0, 0, 0], [0, 0, 0], [0, 0, 0]])[:,:2]\n",
        "plt.ylim(-2,3)\n",
        "plt.xlim(-2,3)\n",
        "QV = plt.quiver(origin[:,0],origin[:,1], data[:, 0], data[:, 1], color=['black', 'green', 'red'], angles='xy', scale_units='xy', scale=1.)\n",
        "plt.grid()\n",
        "plt.xlabel(\"x-axis\")\n",
        "plt.ylabel(\"y-axis\")\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            " shape of matrix containing u & v - (3, 2)\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aagJ3TmS3wDh"
      },
      "source": [
        "We can also understand it from this explanation for a similar example with dimensions given by the instructor from a lecture here - \n",
        "<img src=\"linear_combination_explained.png\" />\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 297
        },
        "id": "ojZkBmfBpB3t",
        "outputId": "f921d06f-015e-4533-ac8b-d489433614f7"
      },
      "source": [
        "# Taking the 3 vectors from the example in the image and plotting them in 3D space (The units of axes are different than the vectors in the plot) \n",
        "u_vec = np.array([1, -1, 0])\n",
        "v_vec = np.array([0, 1, -1])\n",
        "w_vec = np.array([0, 0, 1])\n",
        "\n",
        "data = np.vstack((u_vec, v_vec, w_vec))             \n",
        "origin = np.array([[0, 0, 0], [0, 0, 0], [0, 0, 0]])\n",
        "fig = plt.figure()\n",
        "ax = fig.add_subplot(111, projection='3d')\n",
        "ax.quiver(origin[:,0],origin[:,1], origin[:,2], data[:, 0], data[:, 1], data[:,2], )\n",
        "ax.set_xlim([-1, 2])\n",
        "ax.set_ylim([-1, 2])\n",
        "ax.set_zlim([-1, 2])\n",
        "plt.grid()\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3tQwrViQ6r72"
      },
      "source": [
        "### Span\n",
        "\n",
        "* A span is a set of all possible combinations of vectors that we can reach with a linear combination of a given pair of vectors\n",
        "*\tThe span of most pairs of 2-D vectors is all vectors in the 2-D space. Except, when they line up in the same direction (i.e if they are collinear) , in which case, their span is a line. \n",
        "\n",
        "i.e span( $\\vec{a}$, $\\vec{b}$) = $\\mathbb{R}^2 $  (all vectors in 2D space) , provided they are not collinear.\n",
        "\n",
        "### Collinearity\n",
        "\n",
        "Collinearity is the case when we have p different predictor variables but some of them are linear combinations of others, so they don't add any other information. 2 collinear vectors / variables will have correlation close to $\\pm1$ and can be detected by their correlation matrix.\n",
        "\n",
        "**Multicollinearity** exists when more than 2 vectors are collinear and any pair of vectors may not necessarily have high correlation.\n",
        "\n",
        "### Linear Independence\n",
        "\n",
        "We say that $v_1, v_2, . . . , v_n $ are linearly independent, if none of them is\n",
        "a linear combination of the others. This is equivalent to saying\n",
        "that $x_1v_1 + x_2v_2 + . . . + x_nv_n = 0 $ implies $ x_1 = x_2 = . . . = x_n = 0$\n",
        "\n",
        "Since collinear vectors can be expressed as linear combinations of each other, they are **linearly dependent.**\n",
        "\n",
        "\n",
        "\n",
        "### Basis\n",
        "A basis is that set of linearly independent vectors that span that space.\n",
        "We call these vectors as basis vectors\n",
        "\n",
        "### Vector space Models in NLP\n",
        "\n",
        "A **Vector space** is a set V of vectors, where two operations - vector addition and scalar multiplication are defined. For E.g. IF two vectors $\\hat u$ & $\\hat v$ are in space $V$, then their sum, $\\hat w = \\hat u + \\hat v $ will also lie in the vector space $V$.\n",
        "\n",
        "A 2D vector space is a set of linearly independent vasis vectors with 2 axes.\n",
        "Each axis represents a dimension in the vector space.\n",
        "\n",
        "Recalling the previous plot of vector $ a = (3,5) = 3\\hat i + 5 \\hat j$ again. This vector is represented on a 2D space with 2 linearly independent basis vectors - X & Y, who also represent the 2 axes as well as the 2 dimenions of the space.\n",
        "\n",
        "3 & 5 here are the x,y components of this vector for representation on the X-Y 2D space.\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 261
        },
        "id": "BsZMK7c8weRX",
        "outputId": "4c69010f-5a42-4a87-ef8f-a52fc431574c"
      },
      "source": [
        "data = np.array([[3, 5]])              # List of x,y coordinates for each vector head\n",
        "origin = np.array([[0, 0]])            # List of x,y coordinates for each vector origin\n",
        "plt.ylim(-2,6)\n",
        "plt.xlim(-2,6)\n",
        "plt.quiver(origin[:,0], origin[:,1], data[:, 0], data[:, 1], color=['black'], angles='xy', scale_units='xy', scale=1.)\n",
        "plt.grid()\n",
        "plt.xlabel(\"x-axis\")\n",
        "plt.ylabel(\"y-axis\")\n",
        "plt.show()"
      ],
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 504x252 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "muAmgifuwj6G"
      },
      "source": [
        "**Vector space model** for words is a representation of text in vector space.\n",
        "Here, each word in a corpus is a linearly independent basis vector and each basis vector represents an axis in the vector space. \n",
        "\n",
        "* This means, each word is orthogonal to other words/axes. \n",
        "* For a corpus of vocabulary $|V|$, $\\mathbb{R}$ will contain $|V|$ axes.\n",
        "* Combination of terms represent documents as points or vectors in this space\n",
        "\n",
        "For 3 words, we'll have a 3D vector model represented like this -\n",
        "\n",
        "<img src=\"VSM_words.png\" />\n",
        "\n",
        "The table above the graph represents the TF-IDF incident matrix.\n",
        "\n",
        "$D_1$ = (0.91, 0, 0.0011) represents a document vector in the 3 axes - good, house, car. Similarly, we have $D_2$ & $D_3$ document vectors.\n",
        "\n",
        "How does representation in vector space help us, though?\n",
        "\n",
        "* One of the common application using this representation is information retrieval for search engines, question answering systems and much more.\n",
        "\n",
        "* By representing the text into vectors, we aim to use vector algebra to extract semantics from the text and use it for different applications like searching documents containing the similar sementics as those contained in a given search query. \n",
        "\n",
        "For eg. For a search token 'buy' , we would want to get all the documents containing different forms of this word - buying, bought and even synonyms of the word 'buy'. Such documents can not be captured from other rudimentary methods representing documents as Binary [incident matrix](https://en.wikipedia.org/wiki/Incidence_matrix).\n",
        "\n",
        "This is achieved through distance metrics like cosine similarity between vectors of document & query, where the documents closer to the query are ranked the highest.\n",
        "\n",
        "* The number of words / vocabulary size can be as huge as in millions eg. Google news corpus is 3 Million, which means as many independent axes/dimensions to represent the vectors. Hence, we want to use the operations in vector space to reduce the number of dimensions and bring words with similar to each other in the same axis.\n",
        "\n",
        "### Dense vectors \n",
        "* The above operations are possible on document vectors which are represented by extending the above vector representation of documents to documents represented as distributed or dense vectors. These representations capture the semantics of the text and also captures the linear combinations of word vectors \n",
        "\n",
        "* The previous vector space model with each word representing a separate dimension results in sparse vectors\n",
        "\n",
        "* Dense vectors captures the context in the vector representation. The dense vector of words are such that words appearing in similar contexts will have similar representations.\n",
        "\n",
        "* These dense vectors are also called as word embeddings or distributed representations.\n",
        "\n",
        "* word2vec is one such framework to **learn** dense vectors of words from large corpus. It has 2 variants - skip-gram and CBOW(continuous bag of words)\n",
        "\n",
        "* Some of the other frameworks and techniques to obtain dense vectors are Global Vectors(GloVe), fastText, ELMo(Embeddings from Language Model) and most recent state of the art Bert based approached to obtain contextualized word embeddings during inference.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uyHOTQQJ8J3K"
      },
      "source": [
        "### References -\n",
        "*\t[Linear Combination & span, Khan Academy](https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/linear-combinations/v/linear-combinations-and-span)\n",
        "\n",
        "*\t[Linear Combination, span & basis vectors, 3Blue1Brown](https://www.youtube.com/watch?v=k7RM-ot2NWY)\n",
        "\n",
        "*\t[Independence, Basis & Dimension, MIT](\n",
        "https://www.youtube.com/watch?v=eeMJg4uI7o0) \n",
        "\n",
        "*\t[Matrices-Introduction to Vectors, MIT](https://math.mit.edu/~gs/linearalgebra/linearalgebra5_1-3.pdf)\n",
        "\n",
        "* [Vector space models for NLP, NPTEL](https://www.youtube.com/watch?v=6Nz88LHOIdo)\n",
        "\n",
        "* [Multicollinearity, Carnegie Mellon University Lecture](http://www.stat.cmu.edu/~cshalizi/mreg/15/lectures/lecture-17.pdf)\n",
        "\n",
        "* An introduction to statistical learning, E-Book by Gareth James, Daniela Witten, Trevor Hastie and Robert Tibshirani. \n",
        "\n",
        "* https://www.maths.tcd.ie/~pete/ma1111/chapter3.pdf\n",
        "\n",
        "* [Vector space & Operators, MIT](https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/lecture-notes/MIT8_05F13_Chap_03.pdf)\n",
        "\n",
        "* http://www.math.niu.edu/~beachy/courses/240/06spring/vectorspace.html\n",
        "\n",
        "* [Vector in 2D, 3D Cartisean coordinates](https://mathinsight.org/vectors_cartesian_coordinates_2d_3d)"
      ]
    }
  ]
}