seminars/seminar2/seminar2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   "source": [
    "## Пример эквивалентности норм\n",
    "\n",
    "Все нормы в конечномерном пространстве эквивалентны. Это утверждение означает, что для любых двух норм найдётся неотрицательные константы $C_1, C_2 \\geq 0$ такие что\n",
    "\n",
    "$$ C_1 \\| \\cdot \\|_{*} \\leq \\| \\cdot \\|_{**} \\leq C_2 \\| \\cdot\\|_*$$\n",
    "\n",
    "Покажем это на конкретных примерах для норм $\\|\\cdot\\|_2, \\|\\cdot\\|_1, \\|\\cdot\\|_{\\infty}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Поиск максимума\n",
    "\n",
    "Для рассмотренных норм можно построить сопряжённые нормы с помощью следующего определения\n",
    "\n",
    "$$ \\|z\\|_* = \\max_{\\|x\\| \\leq 1} (z, x). $$\n",
    "\n",
    "Найдём сопряжённую норму для евклидовой нормы. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Вычисление произведения матриц\n",
    "\n",
    "$$ C = AB \\quad c_{ij} = \\sum_{k=1}^m a_{ik}b_{kj} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numba import jit\n",
    "\n",
    "def matmul(a, b):\n",
    "    n = a.shape[0]\n",
    "    k = a.shape[1]\n",
    "    m = b.shape[1]  \n",
    "    c = np.zeros((n, m))\n",
    "    for i in range(n):\n",
    "        for j in range(m):\n",
    "            for s in range(k):\n",
    "                c[i, j] += a[i, s] * b[s, j]\n",
    "                \n",
    "    return c\n",
    "\n",
    "@jit(nopython=True)\n",
    "def numba_matmul(a, b):\n",
    "    n = a.shape[0]\n",
    "    k = a.shape[1]\n",
    "    m = b.shape[1]\n",
    "    c = np.zeros((n, m))\n",
    "    for i in range(n):\n",
    "        for j in range(m):\n",
    "            for s in range(k):\n",
    "                c[i, j] += a[i, s] * b[s, j]\n",
    "    return c\n",
    "\n",
    "n = 500\n",
    "a = np.random.randn(n, n)\n",
    "b = np.random.randn(n, n)\n",
    "\n",
    "%timeit numba_matmul(a, b)\n",
    "%timeit numba_matmul_isj(a, b)\n",
    "%timeit np.dot(a, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Приложение: рекомендательные системы"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Что дано?\n",
    "\n",
    "- Матрица пользователь-продукт\n",
    "- Элементы матрицы – рейтинги, если есть\n",
    "- Неполное заполнение"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Как использовать SVD?\n",
    "\n",
    "- Оригинальная работа [*Performance of recommender algorithms on top-n recommendation tasks*](https://dl.acm.org/citation.cfm?id=1864721), Cremonesi, P., Koren, Y., and Turrin, R\n",
    "- Построим вложение пользователей и продуктов в некоторое подпространство $\\mathbb{R}^n$\n",
    "- Для известных предпочтений найдём наиболее близкие продукты в этом подпространстве\n",
    "- Как это вычислительно сделать?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Датасет Movielens с рейтингами фильмов"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "movies = pd.read_csv('movies.gz', index_col='movieid', header=0, encoding='unicode-escape')[['movienm', 'genreid']]\n",
    "ratings = pd.read_csv('ratings.gz', header=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Пример записей"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
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       "\n",
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       "\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>movienm</th>\n",
       "      <th>genreid</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>movieid</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Toy Story (1995)</td>\n",
       "      <td>Animation, Children's, Comedy</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Jumanji (1995)</td>\n",
       "      <td>Adventure, Children's, Fantasy</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Grumpier Old Men (1995)</td>\n",
       "      <td>Comedy, Romance</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Waiting to Exhale (1995)</td>\n",
       "      <td>Comedy, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>Father of the Bride Part II (1995)</td>\n",
       "      <td>Comedy</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                    movienm                         genreid\n",
       "movieid                                                                    \n",
       "1                          Toy Story (1995)   Animation, Children's, Comedy\n",
       "2                            Jumanji (1995)  Adventure, Children's, Fantasy\n",
       "3                   Grumpier Old Men (1995)                 Comedy, Romance\n",
       "4                  Waiting to Exhale (1995)                   Comedy, Drama\n",
       "5        Father of the Bride Part II (1995)                          Comedy"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "movies.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
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       "\n",
       "    .dataframe thead th {\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>userid</th>\n",
       "      <th>movieid</th>\n",
       "      <th>rating</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>1193</td>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>661</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1</td>\n",
       "      <td>914</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1</td>\n",
       "      <td>3408</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1</td>\n",
       "      <td>2355</td>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1000204</th>\n",
       "      <td>6040</td>\n",
       "      <td>1091</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1000205</th>\n",
       "      <td>6040</td>\n",
       "      <td>1094</td>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1000206</th>\n",
       "      <td>6040</td>\n",
       "      <td>562</td>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1000207</th>\n",
       "      <td>6040</td>\n",
       "      <td>1096</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1000208</th>\n",
       "      <td>6040</td>\n",
       "      <td>1097</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>1000209 rows × 3 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "         userid  movieid  rating\n",
       "0             1     1193       5\n",
       "1             1      661       3\n",
       "2             1      914       3\n",
       "3             1     3408       4\n",
       "4             1     2355       5\n",
       "...         ...      ...     ...\n",
       "1000204    6040     1091       1\n",
       "1000205    6040     1094       5\n",
       "1000206    6040      562       5\n",
       "1000207    6040     1096       4\n",
       "1000208    6040     1097       4\n",
       "\n",
       "[1000209 rows x 3 columns]"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ratings"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "userid     6040\n",
       "movieid    3706\n",
       "dtype: int64"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ratings[['userid', 'movieid']].apply(pd.Series.nunique)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Зададим список предпочтений"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
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       "\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>movienm</th>\n",
       "      <th>genreid</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>movieid</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>858</th>\n",
       "      <td>Godfather, The (1972)</td>\n",
       "      <td>Action, Crime, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1221</th>\n",
       "      <td>Godfather: Part II, The (1974)</td>\n",
       "      <td>Action, Crime, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2023</th>\n",
       "      <td>Godfather: Part III, The (1990)</td>\n",
       "      <td>Action, Crime, Drama</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                 movienm               genreid\n",
       "movieid                                                       \n",
       "858                Godfather, The (1972)  Action, Crime, Drama\n",
       "1221      Godfather: Part II, The (1974)  Action, Crime, Drama\n",
       "2023     Godfather: Part III, The (1990)  Action, Crime, Drama"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "best_movies = \"Godfather\"\n",
    "target_movies = movies.loc[movies.movienm.str.contains(best_movies, flags=2)]\n",
    "target_movies"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Выделим индексы выбранных фильмов"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Int64Index([858, 1221, 2023], dtype='int64', name='movieid')\n"
     ]
    }
   ],
   "source": [
    "favorite_movies_ids = target_movies.index\n",
    "print(favorite_movies_ids)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Создадим матрицу с рейтингами "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(6041, 3953)\n"
     ]
    }
   ],
   "source": [
    "data_matrix = spsp.csr_matrix((ratings.rating.values.astype('f8'), \n",
    "                               (ratings.userid.values, ratings.movieid.values)))\n",
    "print(data_matrix.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Вычислим SVD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "rank = 50\n",
    "_, S, Vt = spsplin.svds(data_matrix, k=rank, return_singular_vectors='vh')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Посмотрим на убывание сингулярных чисел"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fd5d1a3dd10>]"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(S[::-1] / S[-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Получим корреляции"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(50, 3)\n",
      "(3953, 3)\n",
      "(3953,)\n"
     ]
    }
   ],
   "source": [
    "print(Vt[:, favorite_movies_ids].shape)\n",
    "correlation = Vt.T @ Vt[:, favorite_movies_ids]\n",
    "print(correlation.shape)\n",
    "score = correlation.sum(axis=1)\n",
    "print(score.shape)\n",
    "# movies.loc[np.argsort(-Vt.T @ Vt[:, favorite_movies_ids].sum(axis=1))[:15]] # assuming binary preference vector"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Вычислим итоговые рекомендации"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.8412151  0.72298478 0.20254655 0.12884495 0.12727929 0.11651275\n",
      " 0.10928934 0.10901711 0.09848998 0.09220464 0.08291223 0.07980152\n",
      " 0.07433467 0.0699485  0.06720786 0.06554852 0.05886816 0.05670393\n",
      " 0.0564577  0.0531071 ]\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>movienm</th>\n",
       "      <th>genreid</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>movieid</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>858</th>\n",
       "      <td>Godfather, The (1972)</td>\n",
       "      <td>Action, Crime, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1221</th>\n",
       "      <td>Godfather: Part II, The (1974)</td>\n",
       "      <td>Action, Crime, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2023</th>\n",
       "      <td>Godfather: Part III, The (1990)</td>\n",
       "      <td>Action, Crime, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1213</th>\n",
       "      <td>GoodFellas (1990)</td>\n",
       "      <td>Crime, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1953</th>\n",
       "      <td>French Connection, The (1971)</td>\n",
       "      <td>Action, Crime, Drama, Thriller</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>912</th>\n",
       "      <td>Casablanca (1942)</td>\n",
       "      <td>Drama, Romance, War</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>111</th>\n",
       "      <td>Taxi Driver (1976)</td>\n",
       "      <td>Drama, Thriller</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>923</th>\n",
       "      <td>Citizen Kane (1941)</td>\n",
       "      <td>Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2019</th>\n",
       "      <td>Seven Samurai (The Magnificent Seven) (Shichin...</td>\n",
       "      <td>Action, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2194</th>\n",
       "      <td>Untouchables, The (1987)</td>\n",
       "      <td>Action, Crime, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1222</th>\n",
       "      <td>Full Metal Jacket (1987)</td>\n",
       "      <td>Action, Drama, War</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1954</th>\n",
       "      <td>Rocky (1976)</td>\n",
       "      <td>Action, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2872</th>\n",
       "      <td>Excalibur (1981)</td>\n",
       "      <td>Action, Drama, Fantasy, Romance</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3578</th>\n",
       "      <td>Gladiator (2000)</td>\n",
       "      <td>Action, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1466</th>\n",
       "      <td>Donnie Brasco (1997)</td>\n",
       "      <td>Crime, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3362</th>\n",
       "      <td>Dog Day Afternoon (1975)</td>\n",
       "      <td>Comedy, Crime, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2324</th>\n",
       "      <td>Life Is Beautiful (La Vita è bella) (1997)</td>\n",
       "      <td>Comedy, Drama</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1220</th>\n",
       "      <td>Blues Brothers, The (1980)</td>\n",
       "      <td>Action, Comedy, Musical</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1242</th>\n",
       "      <td>Glory (1989)</td>\n",
       "      <td>Action, Drama, War</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1193</th>\n",
       "      <td>One Flew Over the Cuckoo's Nest (1975)</td>\n",
       "      <td>Drama</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                                   movienm  \\\n",
       "movieid                                                      \n",
       "858                                  Godfather, The (1972)   \n",
       "1221                        Godfather: Part II, The (1974)   \n",
       "2023                       Godfather: Part III, The (1990)   \n",
       "1213                                     GoodFellas (1990)   \n",
       "1953                         French Connection, The (1971)   \n",
       "912                                      Casablanca (1942)   \n",
       "111                                     Taxi Driver (1976)   \n",
       "923                                    Citizen Kane (1941)   \n",
       "2019     Seven Samurai (The Magnificent Seven) (Shichin...   \n",
       "2194                              Untouchables, The (1987)   \n",
       "1222                              Full Metal Jacket (1987)   \n",
       "1954                                          Rocky (1976)   \n",
       "2872                                      Excalibur (1981)   \n",
       "3578                                      Gladiator (2000)   \n",
       "1466                                  Donnie Brasco (1997)   \n",
       "3362                              Dog Day Afternoon (1975)   \n",
       "2324            Life Is Beautiful (La Vita è bella) (1997)   \n",
       "1220                            Blues Brothers, The (1980)   \n",
       "1242                                          Glory (1989)   \n",
       "1193                One Flew Over the Cuckoo's Nest (1975)   \n",
       "\n",
       "                                 genreid  \n",
       "movieid                                   \n",
       "858                 Action, Crime, Drama  \n",
       "1221                Action, Crime, Drama  \n",
       "2023                Action, Crime, Drama  \n",
       "1213                        Crime, Drama  \n",
       "1953      Action, Crime, Drama, Thriller  \n",
       "912                  Drama, Romance, War  \n",
       "111                      Drama, Thriller  \n",
       "923                                Drama  \n",
       "2019                       Action, Drama  \n",
       "2194                Action, Crime, Drama  \n",
       "1222                  Action, Drama, War  \n",
       "1954                       Action, Drama  \n",
       "2872     Action, Drama, Fantasy, Romance  \n",
       "3578                       Action, Drama  \n",
       "1466                        Crime, Drama  \n",
       "3362                Comedy, Crime, Drama  \n",
       "2324                       Comedy, Drama  \n",
       "1220             Action, Comedy, Musical  \n",
       "1242                  Action, Drama, War  \n",
       "1193                               Drama  "
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "top_n = 20\n",
    "print(np.sort(score)[::-1][:top_n])\n",
    "movies.loc[np.argsort(-score)[:top_n]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "### Обобщения \n",
    "\n",
    "- Учёт контекста позволяет внести в скалярное произведение матрицу похожести. Получим гибридную модель SVD, подробности в [этой работе](https://arxiv.org/pdf/1802.06398.pdf)\n",
    "- Обзор использования нейронных сетей в рекомендательных системах можно найти [тут](https://arxiv.org/pdf/1707.07435.pdf): неочевидные преимущества по сравнению с моделями, основанными на матричных разложениях"
   ]
  }
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