Merge pull request '2nd meeting' (#2) from 2nd_meeting into master

Reviewed-on: robkw/introduction_to_recommender_systems#2

P4. Appendix - embeddings in high demensional spaces.ipynb

@@ -0,0 +1,96 @@
+{
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+  {
+   "cell_type": "code",
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+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "['dimensions: 1, cases when observation is the nearest: 0.0%',\n",
+       " 'dimensions: 2, cases when observation is the nearest: 0.0%',\n",
+       " 'dimensions: 3, cases when observation is the nearest: 0.0%',\n",
+       " 'dimensions: 10, cases when observation is the nearest: 13.0%',\n",
+       " 'dimensions: 20, cases when observation is the nearest: 61.0%',\n",
+       " 'dimensions: 30, cases when observation is the nearest: 96.0%',\n",
+       " 'dimensions: 40, cases when observation is the nearest: 98.0%',\n",
+       " 'dimensions: 50, cases when observation is the nearest: 100.0%',\n",
+       " 'dimensions: 60, cases when observation is the nearest: 100.0%',\n",
+       " 'dimensions: 70, cases when observation is the nearest: 100.0%',\n",
+       " 'dimensions: 80, cases when observation is the nearest: 100.0%',\n",
+       " 'dimensions: 90, cases when observation is the nearest: 100.0%']"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import random\n",
+    "from numpy.linalg import norm\n",
+    "\n",
+    "dimensions = [1, 2, 3] + [10 * i for i in range(1, 10)]\n",
+    "nb_vectors = 10000\n",
+    "trials = 100\n",
+    "k = 1  # by setting k=1 we want to check how often the closest vector to the avarage of 2 random vectors is one of these 2 vectors\n",
+    "\n",
+    "result = []\n",
+    "for dimension in dimensions:\n",
+    "    vectors = np.random.normal(0, 1, size=(nb_vectors, dimension))\n",
+    "    successes = 0\n",
+    "    for i in range(trials):\n",
+    "        i1, i2 = random.sample(range(nb_vectors), 2)\n",
+    "        target = (vectors[i1] + vectors[i2]) / 2\n",
+    "\n",
+    "        distances = pd.DataFrame(\n",
+    "            enumerate(\n",
+    "                np.dot(target, vectors.transpose())\n",
+    "                / norm(target)\n",
+    "                / norm(vectors.transpose(), axis=0)\n",
+    "            )\n",
+    "        )\n",
+    "        distances = distances.sort_values(by=[1], ascending=False)\n",
+    "        if (i1 in (list(distances[0][:k]))) | (i2 in (list(distances[0][:k]))):\n",
+    "            successes += 1\n",
+    "    result.append(successes / trials)\n",
+    "\n",
+    "[\n",
+    "    f\"dimensions: {i}, cases when observation is the nearest: {100*round(j,3)}%\"\n",
+    "    for i, j in zip(dimensions, result)\n",
+    "]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
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