introduction_to_recommender_systems/P4. Appendix - embeddings in high demensional spaces.ipynb

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       "['dimensions: 1, cases when observation is the nearest: 0.0%',\n",
       " 'dimensions: 2, cases when observation is the nearest: 1.0%',\n",
       " 'dimensions: 3, cases when observation is the nearest: 1.0%',\n",
       " 'dimensions: 10, cases when observation is the nearest: 8.0%',\n",
       " 'dimensions: 20, cases when observation is the nearest: 68.0%',\n",
       " 'dimensions: 30, cases when observation is the nearest: 93.0%',\n",
       " 'dimensions: 40, cases when observation is the nearest: 100.0%',\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%']"
      ]
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     "execution_count": 1,
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   "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",
    "        similarities = pd.DataFrame(\n",
    "            enumerate(\n",
    "                np.dot(target, vectors.transpose())\n",
    "                / norm(target)\n",
    "                / norm(vectors.transpose(), axis=0)\n",
    "            )\n",
    "        )\n",
    "        similarities = similarities.sort_values(by=[1], ascending=False)\n",
    "        if (i1 in (list(similarities[0][:k]))) | (i2 in (list(similarities[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",
    "]"
   ]
  }
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