random_projection.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "6d1990f8-1e95-4b17-b59c-f8940104099c",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.spatial import distance_matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "96922116-43b7-436e-bd37-278a00d9966d",
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 100000\n",
    "N = 200"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "e7866648-c2c6-4e46-80fd-8d9006adb650",
   "metadata": {},
   "outputs": [],
   "source": [
    "epsilon = 0.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "ce08cab8-09f2-495e-b8e9-57e36181b12b",
   "metadata": {},
   "outputs": [],
   "source": [
    "m = int(np.ceil(24 * np.log(N) * (epsilon ** -2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c84c6899-fc8b-41a2-915c-486bce644f1e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3179"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "b9871936-a5a9-4aa5-9efb-fffd3832f0a6",
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.random.normal(size=(N, n)) + 10."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "020790a8-d3cc-4c41-9655-a58a06ef9880",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(200, 100000)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "1ce74833-56cd-4f0c-9f4c-ef1d047fa4e7",
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.random.normal(0, 1/np.sqrt(m), (m, n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "a0c63ce7-a46c-4f36-9249-a92c4e812b22",
   "metadata": {},
   "outputs": [],
   "source": [
    "X_dimred = (A @ X.T).T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "5e14e228-280e-44ea-8f93-2d66ae8db242",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(200, 3179)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_dimred.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "9fc366a2-354b-4649-b612-5490f8a3a3f7",
   "metadata": {},
   "outputs": [],
   "source": [
    "dist_orig = distance_matrix(X, X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "faf8101f-2705-454f-ae29-00e0ffc5ab95",
   "metadata": {},
   "outputs": [],
   "source": [
    "dist_orig = dist_orig[np.triu_indices_from(dist_orig, k=1)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "27d3c0f7-504e-4144-b6ca-6874814612bd",
   "metadata": {},
   "outputs": [],
   "source": [
    "dist_dimred = distance_matrix(X_dimred, X_dimred)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "f0a720a3-e26e-4778-947c-b470c3ac17f6",
   "metadata": {},
   "outputs": [],
   "source": [
    "dist_dimred = dist_dimred[np.triu_indices_from(dist_dimred, k=1)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "c76fa722-1934-4da4-995d-add70f1d9bdc",
   "metadata": {},
   "outputs": [],
   "source": [
    "diff = (dist_dimred ** 2 - dist_orig ** 2) / dist_orig ** 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "3cc1cef9-b3a8-42ca-b773-c81d0da6afbc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.03662259,  0.00642912, -0.00545849, ..., -0.0304041 ,\n",
       "       -0.04691323, -0.02393412])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "820d4ae8-6b9e-4202-a6ad-558667ed4d4b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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V343mxrRfBi4F3ldVt0wqMEmSdGfDnuM+FVhLk7SfTtMRiyRJmqJhTeX7VdWvAbTPcX9+MiFJkqT5DKtx3zaQiE3kkiTNhmE17ocnubGdDrBT+z5AVdWuY49OkiTdwbBhPbcf10GT7AacCDwMKOAlVXXBuI4nSdJKMcrjYONwAvDJqnpOO9b3zlOKQ5KkTpl44k6yK/BE4EUAVXUzcPOk41D3OaiIpG3RKMN6Lrd9gC3AyUm+mOTEJHefQhySJHXONBL3XYD9gXdX1SNpulE9pn+lJOuSbEiyYcuWLZOOUZKkmTSNxH0NcE1VXdi+/zBNIr+DqlpfVWurau2qVasmGqAkSbNq4om7qv4D+FaSfdtZBwJXTDoOSZK6aFp3lb8K+GB7R/nXgBdPKQ5JkjplKom7qjbR9IMuSZK2wjSucUuSpEUycUuS1CEmbkmSOmRaN6dJ0kzo74Fv87GHTCkSaTTWuCVJ6hATtyRJHWJTuTrDQUUkyRq3JEmdYuKWJKlDTNySJHWIiVuSpA7x5jRJ6uFz3Zp11rglSeoQE7ckSR1i4pYkqUNM3JIkdYiJW5KkDvGucs0suziVpDuzxi1JUoeYuCVJ6hATtyRJHWLiliSpQ7w5TZKGWOgmSbtE1aRZ45YkqUNM3JIkdYiJW5KkDjFxS5LUISZuSZI6xMQtSVKHTC1xJ9k+yReT/MO0YpAkqWumWeN+NXDlFI8vSVLnTKUDliR7AYcAfwy8dhoxSNJy6O+gxQ5ZNG7T6jntL4DXA7vMt0KSdcA6gNWrV08mKk2Vw3hK0sIm3lSe5FDguqraOGy9qlpfVWurau2qVasmFJ0kSbNtGte4Hw8cnmQzcDpwQJIPTCEOSZI6Z+KJu6reWFV7VdUa4HnAOVV1xKTjkCSpi3yOW5KkDpnqsJ5VdR5w3jRjkCSpS6xxS5LUISZuSZI6xMQtSVKHmLglSeqQqd6cJkkrnV2iarlZ45YkqUNM3JIkdYhN5ZoYmwy1LXCwHI2bNW5JkjrExC1JUoeYuCVJ6hCvcWtsFrrW57VASdp61rglSeoQE7ckSR1i4pYkqUNM3JIkdYiJW5KkDjFxS5LUISZuSZI6xMQtSVKHmLglSeoQE7ckSR1i4pYkqUNM3JIkdYiDjEjSBPUPrrP52EOmFIm6ysStZeFIX5I0GTaVS5LUISZuSZI6ZOJN5UkeALwfuA9wK7C+qk6YdBySNAu85q2tNY1r3LcA/72qLkqyC7AxydlVdcUUYpEkqVMm3lReVddW1UXt9A+BK4H7TzoOSZK6aKrXuJOsAR4JXDhg2bokG5Js2LJly8RjkyRpFk0tcSf5JeAjwNFVdWP/8qpaX1Vrq2rtqlWrJh+gJEkzaCrPcSfZgSZpf7CqPjqNGLQ0PrctSdMx8Rp3kgDvA66sqndM+viSJHXZNJrKHw/8AXBAkk3t6+ApxCFJUudMvKm8qj4DZNLHlSRpJbDnNEmSOsRBRiSpY+xtbdtm4tZIvItckmaDiVsDmaglaTaZuCVphgz6p9mmcPUycUvSjLMFTL28q1ySpA4xcUuS1CEmbkmSOsTELUlSh5i4JUnqEBO3JEkdYuKWJKlDTNySJHWIHbBIUsc56Mi2xRq3JEkdYo17G7HQf+R2qShJ3WDi3kaZqCWpm0zckrTCeQ18ZfEatyRJHWLiliSpQ2wq7yhvNpOkbZOJW5JWmIX+cfead7fZVC5JUoeYuCVJ6hCbylcIr2lLWiybzrvFxN0RJmZJk2Iin20m7hllopY0q0zs0zWVxJ3kacAJwPbAiVV17DTikCQtzIrEbJl44k6yPfBXwEHANcAXkpxZVVdMOpZJ8osvaaVa6O+bNfLlNY0a96OBr1bV1wCSnA48A+hM4jYJS9LotvZvpol+uGkk7vsD3+p5fw3wG5MMwF7HJGl2jeNv8FL/zi+0/ST/2ZhG4s6AeXWnlZJ1wLr27Y+SXDXCvvcArt/qgI7b2i0mZlHlmVGWZTZZltlkWZbZUv/Ot9vPW5Yx5JEHzrdgGon7GuABPe/3Ar7Tv1JVrQfWb82Ok2yoqrVLC292rKTyWJbZZFlmk2WZTbNSlmn0nPYF4MFJ9k5yV+B5wJlTiEOSpM6ZeI27qm5J8ofAp2geBzupqi6fdBySJHXRVJ7jrqp/Av5pDLveqqb1DlhJ5bEss8myzCbLMptmoiyputN9YZIkaUY5OpgkSR3SicSdZPckZyf5SvvznvOsd1KS65JcNur2Sd6Y5KtJrkry2zNUlqe1MX01yTE9889Isql9bU6yqZ2/JslPe5a9pwNleWuSb/fEfHDPsq6dlz9L8qUklyT5WJLd2vkTOy/zxdazPEn+sl1+SZL9RyjXSJ/LrJQlyQOSnJvkyiSXJ3l1zzbzft9msSztss1JLm3j3dAzv2vnZd+ez31TkhuTHN0um8p5GbE8v5LkgiQ3JXndKNtO5NxU1cy/gD8FjmmnjwGOm2e9JwL7A5eNsj2wH3AxsCOwN3A1sP20y0Jz097VwD7AXdsY9xuw3tuBN7fTa/rLPQvnZVhZgLcCrxuwTefOC/BU4C7t9HE937GJnJdRvjPAwcA/0/Sl8BjgwhHKNdLv3gyV5b7A/u30LsCXF/q+zWpZ2mWbgT0W832dtbL07ec/gAdO67xsRXn2BH4d+OPeGKf9O9OJGjdNl6inttOnAr8zaKWqOh+4YSu2fwZwelXdVFVfB75K0yXrOI1Sltu6ha2qm4G5bmFvkyTAc4HTxhfqgpalLPPst1PnparOqqpb2vU+R9M/wSSN8jk/A3h/NT4H7JbkvgtsO9Lv3jJbdFmq6tqqugigqn4IXEnTW+O0LOW8DNOp89K3zoHA1VX1jfGHPNSC5amq66rqC8DPt2LbsZ+briTue1fVtQDtzz2XaftB3a+O+5d8lLKMEtcTgO9W1Vd65u2d5ItJ/jXJE5Yz6HksR1n+sG1SO6mnSanL5wXgJTS1jjmTOC+jxDbfOsO2Xerv3mIspSy3SbIGeCRwYc/sQd+3cVpqWQo4K8nGNL1JzunseaHpu6O/wjHp8wJL+zsz1d+ZmRmPO8m/APcZsOhN4zzsgHlLvs1+GcoySlzP545f/muB1VX1n0keBXw8yUOr6sYRjzk4kPGW5d3A29r3b6Np+n/JAtss2iTOS5I3AbcAH2xnjeW8LCa2IeuM5fNegqWUpVmY/BLwEeDons96vu/bOC21LI+vqu8k2RM4O8mX2pbFaViO83JX4HDgjT3Lp3FeYGnf+6n+zsxM4q6qp8y3LMl355rB2maX67Zy9/NtP1L3q1trGcoyNK4kdwGeBTyq55g3ATe10xuTXA08BNjAEoyzLFX13Z59vRf4h4W2WYoJnJcjgUOBA6u9wDWu87K1sS2wzl2HbLvU373FWEpZSLIDTdL+YFV9dG6FId+3cVpSWapq7ud1ST5G00R7Ph08L62nAxf1nospnRdY2t+ZYduO/dx0pan8TODIdvpI4BPLtP2ZwPOS7Jhkb+DBwOeXGOtiY+m1ULewTwG+VFXXzM1IsirNWOck2YemLF8bQ/y9llSWvmtfzwTmngbo3HlJ8jTgDcDhVfWTuQ0meF5G6Ur4TOCFaTwG+EHblDds26X+7i3GosvS3vvxPuDKqnpH7wZDvm/jtJSy3D3JLm3sd6e5AbL3d6Qz56VneX9L4bTOCyyt++3p/s4s551u43oB9wI+DXyl/bl7O/9+wD/1rHcaTdPkz2n+I3rpsO3bZW+iuTvwKuDpM1SWg2nuiL0aeFPfPk4BXt4379nA5TR3N14EHDbrZQH+BrgUuITmy37frp4XmhvovgVsal/vmfR5GRQb8PK57wpN895ftcsvBdaOUK55f3fGfD4WVRbgN2maLC/pORcHL/R9m9Gy7NN+by5uv0OdPS/tsp2B/wTu0bfPqZyXEctzH5pcciPw/XZ612n/zthzmiRJHdKVpnJJkoSJW5KkTjFxS5LUISZuSZI6xMQtSVKHmLg1VknuldtH/fmP3D4K0PeTXDGB478oybsWWOdJSR7X8/7lSV44hlh+1P68X5IP98w/LU13j69JMxrRpjRdpD5ouWMYpyRPSDMi16YkOy1xX/87ybwd5mzFfpb9XCY5Jclz2ukTk+w3ZN0XJbnfch5fmpme07QyVdV/Ao+AZvg+4EdVdXyafqSXpYekJHep2wf4WIwnAT8C/h2gqsY6JGo1vWHN/eG/D/C4qnpg+/4Y4BNV9ZZR95dk+6r6xViC3TovAI6vqpOXuqOqevOg+Vtb1gmcy5ctsMqLaDoUWXLPf9Ica9yapu2TvLetpZ01V0tL8qAkn0wzsMK/JfmV/g3TjOG7PslZwPvbHso+kuQL7evxA7Y5LMmFbW32X5Lcu/0H4uXAa9qa4hPafb8uya8m+XzP9muSXNJOPyrNoCEbk3wqA0ZzantVuqCN5219+5nrHeosYM/22G8BjgZeluTcdt0jkny+Xf7/cnsvbD9qa6UXAo9dYL0/TnJxks8luXc7/95pxg2/uH09btjx+sp1YPsZXppmUIgdk7yMZrS6Nyf5YN/6a9KMVX5q27Lw4SQ7t8ve3H4+l7XnM+383lrt5na9zwC/27PfPZNsbKcfnqSSrG7fX51k57lz2c47KskVbQynt/Pu3pbhC22Z7jRyXRrvarf9R3oGjUhyXpK1SbZvY76s/Vxe08a/Fvhg+3nuNKS85yU5rv3sv5x2MJp2v8e3+7wkyauGff8GlVEr0KR6qPHli55xd2nGqb4FeET7/kPAEe30p4EHt9O/AZwzz742Aju17/8W+M12ejVNl5fQ1Hje1U7fE27rdOhlwNv74xoQ5yZgn3b6DcAfATvQ1M5XtfN/DzhpQIxnAi9sp19J09owV/bL+qcHHPtXgb8Hdmjf/3XP/gp47ojrHdZO/ynwR+30GTQDcEAztvA9hu2nJ7670fQQ95D2/ft79nMK8JwBn8OaNo7Ht+9P6iljby+Gf9MT6237ohmT+vXzfKcuB3YF/pCmG8oXAA8ELhjweX4H2LGd3q39+Sfc/r3bjaYnrLv3HeNZwNnt53Q/mh605mI7jyY5Pwo4u2eb3XqX98yfr7zncfv38WDgX9rpV9D0uz431vvuDPn+DSqjr5X3sqlc0/T1qtrUTm8E1qQZ1elxwN+1lRGAHefZ/syq+mk7/RRgv55tdk3bz3OPvYAz2trJXYGvjxDjh2hqksfS/IH8PWBf4GE0ozVB8wf92gHbPp6my1No/kgfN8Lxeh1IkxC+0B5nJ24fsOAXNH/QF1rvZm6/JLEROKidPgB4IUA1Tc8/SPIHQ/YzZ1+a8/bl9v2pNP+U/MUCZflWVX22nf4AcBRwPPDkJK+n6Q5zd5pE/PcDtj9jnv3+O83n/ESaJPw0mm43/23AupfQ1H4/Dny8nfdU4PC5WjnNPyaracbxnvNE4LT2c/pOknMG7PtrwD5J3gn8I01LyiDDyjs3IMpGmn92oPlev6faS0FVdUOShzH/929QGbXCmLg1TTf1TP+CJlFsB3y/qh4xwvY/7pneDnhsTyIHoCeRA7wTeEdVnZnkSTS1sYWcQfNPxEeBqqqvJPk14PKqeuwI2y+lT+EAp1bVGwcs+1ndfq132Ho/r6q5GH7B8N/5YfvpXWcx+j+HSnI3mlr92qr6Vpp7IO42z/Y/BkhyMs0Y29+pqoNpEvQTaGrZn6BpFSkG3z9xCE0SPhz4n0ke2pbn2VV11VbGf8eFVd9L8nDgt2n+kXkufUNTjlDeud+H3vOUAccO83//7lTGWtr9H5pBXuPWTKlm7OSvJ/lduO364sNH2PQsmuZS2u0eMWCdewDfbqeP7Jn/Q6C/dj4Xz9U0f0j/J7fX+q4CViV5bHusHdok0O+zNKMGQdOEu7U+DTwnzVjMJNk9yQOXsF7/Nq9o198+ya4j7udLNC0jv9y+/wPgX0coy+q5z4tmhKjPcHvSur5taXnOQjupqhdX1SPapA3NEJdHAF+pqluBG2iamj/bu12S7YAHVNW5wOtpmsV/CfgU8Kqea82PHHDY82lGq9u+ba15cv8KSfYAtquqj9B8V/ZvF/V+t7a6vDTf65enGcqXJLszz/dvSBm1wpi4NYteALw0ydyoSHe6YWiAo4C17U05V9DccNbvrTS1538Dru+Z//fAM9sbiJ4wYLszaJLDhwCq6maaP7rHtTFuomne7/dq4JVJvkDzT8NWqaoraK6pn5XmprizgTvdBDfqegNie3KSS2maZh86yn6q6mfAi2k+x0uBW4FR7ty+Ejiy3e/uwLur6vvAe2lGhvo4zTXqrVJVm9vJ89ufn6Fpsfle36rbAx9oY/4i8Oft8d9Gc834kjQ3DL6NO/sYzUhPlwLvZvA/KvcHzkuyieb6/FyrxSnAe9r5N7H15T0R+GYb38XA7w/5/s1XRq0wjg4maazSPvpXVQ+bdizSSmCNW5KkDrHGLUlSh1jjliSpQ0zckiR1iIlbkqQOMXFLktQhJm5JkjrExC1JUof8fy3XbOCeVHz7AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8,6))\n",
    "plt.hist(diff, bins=100, density = True)\n",
    "plt.xlabel('The relative difference of pair-wise distances')\n",
    "plt.ylabel('Probability Density')\n",
    "plt.savefig('img/random_proj.jpg')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b983aa7e-e0a8-4aaf-ae40-181503035233",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.9.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}