ds_math/normalize_vs_cosine.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "\n",
    "import numpy as np\n",
    "from sklearn import preprocessing\n",
    "from importlib import reload\n",
    "from sklearn.metrics.pairwise import cosine_similarity, linear_kernel"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### normalize in sklearn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.41, -0.41,  0.82],\n",
       "       [ 1.  ,  0.  ,  0.  ],\n",
       "       [ 0.  ,  0.71, -0.71]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = [[ 1., -1.,  2.],\n",
    "     [ 2.,  0.,  0.],\n",
    "     [ 0.,  1., -1.]]\n",
    "X_normalized = preprocessing.normalize(X, norm='l2')\n",
    "\n",
    "(X_normalized).round(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1., 1., 1.])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.norm(X_normalized, axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.41, -0.41,  0.82],\n",
       "       [ 1.  ,  0.  ,  0.  ],\n",
       "       [ 0.  ,  0.71, -0.71]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_normalized = preprocessing.normalize(X, norm='l2', axis=1)\n",
    "(X_normalized).round(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.45, -0.71,  0.89],\n",
       "       [ 0.89,  0.  ,  0.  ],\n",
       "       [ 0.  ,  0.71, -0.45]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_normalized = preprocessing.normalize(X, norm='l2', axis=0)\n",
    "(X_normalized).round(2)                                   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1., 1., 1.])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.norm(X_normalized, axis=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7f054b217128>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_samples = 100\n",
    "n_features = 2\n",
    "X = np.random.randn(n_samples, n_features)\n",
    "plt.figure(figsize=(5, 5))\n",
    "plt.scatter(X[:,0], X[:,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,\n",
       "       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,\n",
       "       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,\n",
       "       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,\n",
       "       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,\n",
       "       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_normalized = preprocessing.normalize(X, norm='l2')\n",
    "np.linalg.norm(X_normalized, axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7f054b1a0828>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(5, 5))\n",
    "plt.scatter(X_normalized[:,0], X_normalized[:,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### cosine similarity in sklearn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.spatial.distance import cosine\n",
    "from sklearn.metrics.pairwise import cosine_similarity\n",
    "from sklearn.neighbors import NearestNeighbors\n",
    "\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.7071067811865475, 0.7071067811865475)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.array([2, 2]).reshape(1, -1)\n",
    "y = np.array([2, 0]).reshape(1, -1)\n",
    "1 - cosine(x, y), cosine_similarity(x, y)[0][0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.7071067811865475, -0.7071067811865475)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.array([2, 2]).reshape(1, -1)\n",
    "y = np.array([-2, 0]).reshape(1, -1)\n",
    "1 - cosine(x, y), cosine_similarity(x, y)[0][0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### normalize v.s. cosine"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.45, 1.78, 0.61, 1.02, 0.87],\n",
       "       [1.81, 1.22, 0.  , 1.91, 1.72],\n",
       "       [0.06, 0.69, 0.72, 0.33, 0.03],\n",
       "       [0.34, 0.77, 1.19, 0.4 , 0.69],\n",
       "       [1.24, 1.43, 1.99, 0.46, 0.52],\n",
       "       [0.72, 1.66, 0.66, 1.08, 0.64],\n",
       "       [0.02, 0.19, 1.97, 1.22, 1.99],\n",
       "       [1.53, 0.9 , 1.16, 1.38, 0.24]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_samples = 8\n",
    "n_features = 5\n",
    "X = np.random.uniform(0, 2, size=(n_samples, n_features))\n",
    "(X).round(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.03,  2.61,  1.67,  1.56, -0.5 ],\n",
       "       [ 2.2 ,  2.89,  2.77,  2.64, -0.9 ],\n",
       "       [-0.61,  2.38,  0.55,  0.24,  0.71],\n",
       "       [ 2.15,  0.11,  1.81,  1.76,  0.43],\n",
       "       [-0.82,  1.11, -0.76,  1.92,  0.55],\n",
       "       [ 1.22,  1.07,  2.  ,  0.5 ,  2.3 ],\n",
       "       [ 1.57,  2.43,  2.54,  1.39,  0.84],\n",
       "       [ 0.02,  1.13,  2.16, -0.55, -0.93]])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y = np.random.uniform(-1, 3, size=(n_samples, n_features))\n",
    "(Y).round(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.72,  0.82,  0.63,  0.72,  0.4 ,  0.78,  0.9 ,  0.27],\n",
       "       [ 0.45,  0.61,  0.39,  0.72,  0.52,  0.72,  0.7 , -0.14],\n",
       "       [ 0.95,  0.88,  0.76,  0.6 ,  0.31,  0.67,  0.93,  0.76],\n",
       "       [ 0.74,  0.76,  0.66,  0.71,  0.2 ,  0.93,  0.95,  0.59],\n",
       "       [ 0.76,  0.87,  0.56,  0.79,  0.03,  0.85,  0.98,  0.69],\n",
       "       [ 0.85,  0.85,  0.76,  0.66,  0.54,  0.75,  0.93,  0.35],\n",
       "       [ 0.44,  0.46,  0.4 ,  0.65,  0.27,  0.88,  0.69,  0.24],\n",
       "       [ 0.71,  0.93,  0.35,  0.94,  0.25,  0.72,  0.91,  0.38]])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cosine_similarity(X, Y)).round(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(np.dot(X, Y.T), cosine_similarity(X, Y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1., 1., 1., 1., 1., 1., 1., 1.])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_normalized = preprocessing.normalize(X, norm='l2')\n",
    "Y_normalized = preprocessing.normalize(Y, norm='l2')\n",
    "\n",
    "np.linalg.norm(X_normalized, axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.72,  0.82,  0.63,  0.72,  0.4 ,  0.78,  0.9 ,  0.27],\n",
       "       [ 0.45,  0.61,  0.39,  0.72,  0.52,  0.72,  0.7 , -0.14],\n",
       "       [ 0.95,  0.88,  0.76,  0.6 ,  0.31,  0.67,  0.93,  0.76],\n",
       "       [ 0.74,  0.76,  0.66,  0.71,  0.2 ,  0.93,  0.95,  0.59],\n",
       "       [ 0.76,  0.87,  0.56,  0.79,  0.03,  0.85,  0.98,  0.69],\n",
       "       [ 0.85,  0.85,  0.76,  0.66,  0.54,  0.75,  0.93,  0.35],\n",
       "       [ 0.44,  0.46,  0.4 ,  0.65,  0.27,  0.88,  0.69,  0.24],\n",
       "       [ 0.71,  0.93,  0.35,  0.94,  0.25,  0.72,  0.91,  0.38]])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cosine_similarity(X_normalized, Y_normalized).round(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(np.dot(X, Y.T), cosine_similarity(X, Y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(cosine_similarity(X, Y), cosine_similarity(X_normalized, Y_normalized))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(np.dot(X_normalized, Y_normalized.T), cosine_similarity(X_normalized, Y_normalized))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "http://scikit-learn.org/stable/modules/metrics.html#cosine-similarity\n",
    "normalized vectors, in which case cosine_similarity is equivalent to linear_kernel, only slower."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(linear_kernel(X_normalized, Y_normalized), cosine_similarity(X_normalized, Y_normalized))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So the euclidean distance will degrade to  $\\sqrt{2 − 2x^T y}$, i.e., $\\sqrt{2 - 2 \\times cosinesimilarity}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.7509789525594854, 0.7509789525594854)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sqrt(sum((X_normalized[0] - Y_normalized[0])**2)), np.linalg.norm(X_normalized[0] - Y_normalized[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.7509789525594853, 0.7509789525594854)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sqrt(2 - 2 * np.dot(X_normalized[0], Y_normalized[0])), np.linalg.norm(X_normalized[0] - Y_normalized[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.75097895]])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sqrt(2 - 2 * cosine_similarity(X_normalized[0].reshape(1, -1), Y_normalized[0].reshape(1, -1)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "http://scikit-learn.org/stable/modules/generated/sklearn.neighbors.KNeighborsClassifier.html\n",
    "metric : string or callable, default ‘minkowski’\n",
    "the distance metric to use for the tree. The default metric is minkowski, and with p=2 is equivalent to the standard Euclidean metric. See the documentation of the DistanceMetric class for a list of available metrics."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- https://stackoverflow.com/questions/34144632/using-cosine-distance-with-scikit-learn-kneighborsclassifier\n",
    "you can get the same ordering as the cosine distance by normalizing your data and then using the euclidean distance. So long as you use the uniform weights option, the results will be identical to having used a correct Cosine Distance.\n",
    "\n",
    "\n",
    "- https://github.com/stanfordnlp/GloVe/blob/765074642a6544e47849bb85d8dc2e11e44c2922/eval/python/evaluate.py#L75-L76\n",
    "```\n",
    "#cosine similarity if input W has been normalized\n",
    "dist = np.dot(W, pred_vec.T)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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