Norm

Norm.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Norm (L1, L2 and L∞)\n",
    "- https://mathtrain.jp/lpnorm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "ノルムとは、ベクトルの大きさを表す量のことである。<br>\n",
       "n次元ベクトル$$x=(x_1,x_2,...,x_n)$$を考える。一般に、$$1 \\leq p < \\infty$$ に対して、\n",
       "以下をxのLPノルムと呼び、$$||x||_p$$と書く。<br><br>\n",
       "$$||x||_p = \\sqrt[p]{|x_1|^p+|x_2|^p+...+|x_n|^p}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%latex\n",
    "ノルムとは、ベクトルの大きさを表す量のことである。<br>\n",
    "n次元ベクトル$$x=(x_1,x_2,...,x_n)$$を考える。一般に、$$1 \\leq p < \\infty$$ に対して、\n",
    "以下をxのLPノルムと呼び、$$||x||_p$$と書く。<br><br>\n",
    "$$||x||_p = \\sqrt[p]{|x_1|^p+|x_2|^p+...+|x_n|^p}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**L2ノルム**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "特にpの表記無く$$||x||$$と記載される場合、一般にL2ノルムを指す。<br>\n",
       "これは普通の意味での\"長さ\"であり、ユークリッドノルムとも呼ばれる。<br><br>\n",
       "$$||x||_2 = \\sqrt{x_1^2+x_2^2+...+x_n^2}$$"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%latex\n",
    "特にpの表記無く$$||x||$$と記載される場合、一般にL2ノルムを指す。<br>\n",
    "これは普通の意味での\"長さ\"であり、ユークリッドノルムとも呼ばれる。<br><br>\n",
    "$$||x||_2 = \\sqrt{x_1^2+x_2^2+...+x_n^2}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**L1ノルム**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "L1ノルムは、各成分の絶対値の和である。ユークリッド距離に対して、こちらをマンハッタン距離という。<br><br>\n",
       "$$||x||_1 = |x_1|+|x_2|+...+|x_n|$$"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%latex\n",
    "L1ノルムは、各成分の絶対値の和である。ユークリッド距離に対して、こちらをマンハッタン距離という。<br><br>\n",
    "$$||x||_1 = |x_1|+|x_2|+...+|x_n|$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**L∞**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$x_1$$ ~ $$x_n$$の中で、最も絶対値が大きいものの一つを$$x_k$$とする。\n",
       "するとpが十分大きい時、以下が成り立つ。<br><br>\n",
       "$$\\sqrt[p]{|x_1|^p+|x_2|^p+...+|x_n|^p} \\simeq |x_k|$$<br><br>\n",
       "そこで、L∞ノルムは絶対値最大の成分の絶対値と定義される。無限大ノルム，supノルムなどとも呼ばれる。"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%latex\n",
    "$$x_1$$ ~ $$x_n$$の中で、最も絶対値が大きいものの一つを$$x_k$$とする。\n",
    "するとpが十分大きい時、以下が成り立つ。<br><br>\n",
    "$$\\sqrt[p]{|x_1|^p+|x_2|^p+...+|x_n|^p} \\simeq |x_k|$$<br><br>\n",
    "そこで、L∞ノルムは絶対値最大の成分の絶対値と定義される。無限大ノルム，supノルムなどとも呼ばれる。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**単位円の図示**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "二次元ベクトルに対して$$||x||_p=1$$であるような領域（単位円）を図示する。\n",
       "pを1から徐々に増やしていくにつれて単位円はふくらんでいく。(n≥3でも同様)"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%latex\n",
    "二次元ベクトルに対して$$||x||_p=1$$であるような領域（単位円）を図示する。\n",
    "pを1から徐々に増やしていくにつれて単位円はふくらんでいく。(n≥3でも同様)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import math\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "横軸が変数x_1、縦軸が変数x_2を表すとする。\n",
      "青がL1ノルム、緑がL2ノルム、赤がL∞ノルムの単位円である。\n"
     ]
    },
    {
     "data": {
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x6qtVjOs/Tuch04hCMstVrFCRCQMnsHLnSv7y778kpc1M/+DeVATkhI8n8PRH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\nFJIiIh4UkiIiHv4/tBvb8/B5meMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1089b5a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"横軸が変数x_1、縦軸が変数x_2を表すとする。\")\n",
    "print(\"青がL1ノルム、緑がL2ノルム、赤がL∞ノルムの単位円である。\")\n",
    "fig, ax = plt.subplots(1,1,figsize=(5,5))\n",
    "ax.set_title('Norm')\n",
    "# set lim\n",
    "xlim = ax.set_xlim(-2,2)\n",
    "ylim = ax.set_ylim(-2,2)\n",
    "# draw axis\n",
    "x_axis = ax.plot(np.linspace(-2, 2), [0]*50, c='black')\n",
    "y_axis = ax.plot([0]*50, np.linspace(-2, 2), c='black')\n",
    "# x range\n",
    "x_p = np.linspace(0, 1) \n",
    "x_m = np.linspace(-1, 0) \n",
    "# draw L1 Norm = 1 area\n",
    "plot1 = ax.plot(x_p,1-x_p,c='b')\n",
    "plot2 = ax.plot(x_p,-1+x_p,c='b')\n",
    "plot3 = ax.plot(x_m,1+x_m,c='b')\n",
    "plot4 = ax.plot(x_m,-1-x_m,c='b')\n",
    "# draw L2 Norm = 1 area\n",
    "y1 = list(map(lambda x: math.sqrt(1-x**2),x_p))\n",
    "y2 = list(map(lambda x: -math.sqrt(1-x**2),x_p))\n",
    "y3 = list(map(lambda x: math.sqrt(1-x**2),x_m))\n",
    "y4 = list(map(lambda x: -math.sqrt(1-x**2),x_m))\n",
    "plot5 = ax.plot(x_p,y1,c='g')\n",
    "plot6 = ax.plot(x_p,y2,c='g')\n",
    "plot7 = ax.plot(x_m,y3,c='g')\n",
    "plot8 = ax.plot(x_m,y4,c='g')\n",
    "# draw L∞ Norm = 1 area\n",
    "xrange = np.linspace(-1, 1)\n",
    "yrange = np.linspace(-1, 1)\n",
    "plot9 = ax.plot(xrange,[1]*50,c='r')\n",
    "plot10 = ax.plot(xrange,[-1]*50,c='r')\n",
    "plot11 = ax.plot([1]*50,yrange,c='r')\n",
    "plot12 = ax.plot([-1]*50,yrange,c='r')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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