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{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "232e5515",
   "metadata": {},
   "source": [
    "### Machine Learning (Background): Vector p-norms\n",
    "$\\lVert \\mathbf{v} \\rVert_p=(\\lvert v_1\\rvert^p+\\lvert v_2\\rvert^p+\\ldots+\n",
    "\\lvert v_n\\rvert^p)^\\frac{1}{p}$\n",
    "###### by Hamed Shah-Hosseini\n",
    "Explanation at: https://www.pinterest.com/HamedShahHosseini/Machine-Learning/Background-Knowledge\n",
    "<br>Explanation in Persian: https://www.instagram.com/words.persian\n",
    "<br>Code that: https://github.com/ostad-ai/Machine-Learning"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "id": "b5f0a6f3",
   "metadata": {},
   "outputs": [],
   "source": [
    "# importing modules\n",
    "# درون‌بَری سَنجانه‌ها\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "id": "4a83fab7",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x288 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# showing special p-norms as images\n",
    "# نمایش پ-هنجارهای ویژه به نشان پنداره‌ها\n",
    "x=np.arange(-50,50,.5)\n",
    "y=np.arange(-50,50,.5)\n",
    "X,Y=np.meshgrid(x,y,indexing='ij')\n",
    "points = np.vstack([X.reshape(-1), Y.reshape(-1)]).T\n",
    "plt.figure(figsize=(12,4))\n",
    "ords=[1,2,np.inf]\n",
    "titles=['1-norm','2-norm','inf-norm']\n",
    "for i in range(len(ords)):\n",
    "    plt.subplot(1,3,i+1); plt.title(titles[i])\n",
    "    norms = np.linalg.norm(points, ord=ords[i],axis=1)\n",
    "    norms=norms.reshape(X.shape)\n",
    "    plt.imshow(norms,cmap=plt.cm.Accent,\n",
    "               origin='lower',extent=[-5,5,-5,5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f03b778a",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.8.15"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}