lecture-7.ipynb

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     "source": [
      "Lecture 7: Singular value decomposition"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
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     "source": [
      "## Syllabus\n",
      "**Week 1:** Python intro  \n",
      "**Week 2:** Matrices, vectors, norms, ranks  \n",
      "**Week 3:** Linear systems, eigenvectors, eigenvalues  \n",
      "**Week 4:** Singular value decomposition + test + homework seminar"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
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     "source": [
      "##Recap of the previous lecture\n",
      "- LU decomposition = Solving linear systems\n",
      "- QR decomposition = Solving linear systems (can be more accurate in the ill-conditioned case, slower than LU, used in the orthogonalization) "
     ]
    },
    {
     "cell_type": "markdown",
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     "source": [
      "## Today lecture\n",
      "Today we will talk about the singular value decomposition (SVD)\n",
      "- SVD and best low-rank approximation\n",
      "- Applications "
     ]
    },
    {
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     "source": [
      "## Singular value decomposition: definition\n",
      "Any $n \\times m$ matrix $A$ can be decomposed into a product  \n",
      "$$\n",
      "   A = U \\Sigma V^*,\n",
      "$$\n",
      "where $U$ is an $n \\times k$ unitary matrix, $V$ is an $m \\times k$ unitary matrix, and $\\Sigma = \\mathrm{diag}(\\sigma_1, \\ldots, \\sigma_k)$ a diagonal matrix with $\\sigma_1 \\geq \\ldots \\sigma_k \\geq 0$ and $k = \\min(m, n)$.\n"
     ]
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     "source": [
      "## SVD and low-rank approximation\n",
      "One of the main features of the SVD is to solve the minimization problem  \n",
      "$$\n",
      "  \\Vert A - A_r \\Vert \\rightarrow \\min,\n",
      "$$\n",
      "i.e. find a closest **rank-r** matrix to a given matrix.  \n",
      "The solution is given as  \n",
      "$$\n",
      "     A_r = U_r \\Sigma_r V^*_r,\n",
      "$$\n",
      "where $U_r$ are the first $r$ columns of the matrix $U$, $V_r$ are the first $r$ columns of the matrix $V$ and $\\Sigma_r$ are the first $r$ singular values"
     ]
    },
    {
     "cell_type": "markdown",
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     "source": [
      "## Existence of the SVD\n",
      "**Theorem.** Any matrix has an SVD decomposition."
     ]
    },
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     "source": [
      "**Proof.**  Recall the definition of $\\sigma = \\Vert A \\Vert_2$ as an operator norm. Thus, there exist vectors $x$, $y$ of unit length such that $Ay = \\sigma x$. Now introduce two unitary matrices  \n",
      "$$U = [x, U_1], \\quad V = [y, V_1].$$\n",
      "Then,\n",
      "$$\n",
      "   U^* A V = \\begin{bmatrix}\n",
      "   \\sigma & w^* \\\\\n",
      "   0 & B\n",
      "   \\end{bmatrix},\n",
      "$$\n",
      "and\n",
      "$$\n",
      "   \\Vert U^* A V \\begin{bmatrix}\n",
      "   \\sigma \\\\\n",
      "   w \n",
      "   \\end{bmatrix}\\Vert^2_2 \\geq (\\sigma^2 + w^* w)^2 \\longrightarrow \\Vert U^* A V \\Vert^2_2 \\geq \\sigma^2 + w^* w.\n",
      "$$\n",
      "Since $\\Vert U^* A V \\Vert = \\sigma$, we get $w = 0$. "
     ]
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     "source": [
      "## SVD and eigenvalues\n",
      "From $A = U \\Sigma V^*$ we get\n",
      "$$\n",
      "   A A^* = U \\Sigma^2 U^*, \\quad A^* A = V \\Sigma^2 V^*,\n",
      "$$\n",
      "i.e. you can compute the singular values as a square roots of the eigenvalues of $A^*A$ (or $A A^*$).\n",
      "It is not a good idea, since the maximal precision you might get this way is $\\sqrt{\\varepsilon}$. You just loose half of the digits!"
     ]
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     "source": [
      "## How the SVD is actually computed\n",
      "\n",
      "All SVD computations are done in two steps.\n",
      "\n",
      "1. Reduction of a given matrix to a **bidiagonal form**, $B = U^* A V$ by means of Householder/Givens transformations $\\mathcal{O}(n^3)$\n",
      "2. Solving the SVD problem for a bidiagonal matrix.\n",
      "\n",
      "The Golub-Kahan bidiagonalization algorithm was the first (and still one of the most popular) algorithms for solving the SVD problem. It is basically a QR algorithm, **implicitly** applied to a matrix $B^* B$.  "
     ]
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     "source": [
      "## Some other SVD properties\n",
      "1. SVD can be used to compute **pseudoinverses** of a matrix, trying to solve the ill-conditioned / rectangular systems, \n",
      "  $$A^{\\dagger} = V^* \\Sigma^{\\dagger} U,$$\n",
      "  where $\\Sigma^{\\dagger}_i = \\min(\\frac{1}{\\sigma_i}, \\frac{1}{\\delta})$, i.e. small singular values are  smoothed\n",
      "  \n",
      "2. SVD gives a full information about the **nullspace** and **range** of the matrix"
     ]
    },
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     "cell_type": "markdown",
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     "source": [
      "## Application 1: Latent semantic analysis\n",
      "One of the most famous application is Latent semantic indexing, see, for example,  \n",
      "[Deerwester, Scott C., et al. \"Indexing by latent semantic analysis.\" (1990)](http://www.cob.unt.edu/itds/faculty/evangelopoulos/dsci5910/LSA_Deerwester1990.pdf)  \n",
      "\n",
      "The problem setup: we have a set of text documents $D_1, \\ldots, D_N.$ We want to solve the search problem: i.e., we have a query as a set of words, and we want to find the best documents.  \n",
      "Our data is processed as follows:  for each document we create a list of words contained in the document, and count the frequencies of each word. This is called the **bag of words** model (i.e., the document is unordered set of words).  "
     ]
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     "source": [
      "## Term-document matrix\n",
      "This is how the term-document matrix $A$ is obtained. Its row size is **the size of the dictionary**.   \n",
      "Its column size is **the number of documents**.  An element $A_{ij}$ the the frequency of occurence of the $i$-th word in the $j$-document.\n",
      "\n",
      "To do the search, we can just multiply a term-document matrix by a **search vector**, i.e. a list of words."
     ]
    },
    {
     "cell_type": "markdown",
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     "source": [
      "**The problem**: The document will be returned only if there is an exact word match.  However, we might search for \"Samuel Clemens\", and hope to get the results for \"Mark Twain\" as well.  But there might be no exact match!  \n",
      "How the SVD can help?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
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       "slide_type": "slide"
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     "source": [
      "## Idea of LSI\n",
      "Compute **low-rank approximation** $A_r$ of the term-document matrix $A$.  \n",
      "$$A \\approx A_r,$$\n",
      "and **we do not care about the approximation error** (i.e., we do not require it to be small).  \n",
      "The matrix $A_r$ can be then used to do queries.  \n",
      "\n",
      "We project the documents to **low-dimensional subspace**, given a query $q$ the projection is  \n",
      "$$q_r = \\Sigma^{-1}_r U^{\\top}_r q$$\n",
      "Now we can compute the similarity between $d_r$ and other projected documents  \n",
      "$$\\widehat{d}_i = \\Sigma^{-1}_r U^{\\top}_r d_i,$$\n",
      "and compute the **cosine** of the angles  between the query and the projected document."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
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       "slide_type": "slide"
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     },
     "source": [
      "## Demo\n",
      "Now we can test a demo database"
     ]
    },
    {
     "cell_type": "code",
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     "input": [
      "import numpy as np\n",
      "import pandas as pd\n",
      "import re #Regular expressions\n",
      "rows = ['human', 'interface', 'computer', 'user', 'system', 'response', 'time', 'EPS', 'survey', 'trees', 'graph', 'minors']\n",
      "nterms = len(rows)\n",
      "docs = []\n",
      "docs += ['Human machine interface for Lab ABC computer applications']\n",
      "docs += ['A survey of user opinions of computer system response time']\n",
      "docs += ['The EPS user interfaces management system']\n",
      "docs += ['System and human system engineering testing of EPS']\n",
      "docs += ['Relation of user-perceived response time on user management']\n",
      "docs += ['The generation of random, binary, unordered trees']\n",
      "docs += ['The intersection graph of paths in trees']\n",
      "docs += ['Graph minors IV: Width of trees and well-quasi-ordering']\n",
      "docs += ['Graph minors: A survey']\n",
      "ndocs = len(docs)\n",
      "term_doc = np.zeros((nterms, ndocs))\n",
      "\n",
      "nonletters = re.compile('[^a-zA-Z]+') # split by anything non-alphanumeric\n",
      "docs = [set(nonletters.split(i.lower())) for i in docs]\n",
      "rows = [i.lower() for i in rows]\n",
      "for i in xrange(nterms):\n",
      "    for j in xrange(ndocs):\n",
      "        if rows[i] in docs[j]:\n",
      "            term_doc[i, j] = 1\n",
      "\n",
      "#Use pandas to plot        \n",
      "pd.DataFrame(data=term_doc,index=rows)"
     ],
     "language": "python",
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     "outputs": [
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       "html": [
        "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
        "<table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>0</th>\n",
        "      <th>1</th>\n",
        "      <th>2</th>\n",
        "      <th>3</th>\n",
        "      <th>4</th>\n",
        "      <th>5</th>\n",
        "      <th>6</th>\n",
        "      <th>7</th>\n",
        "      <th>8</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>human</th>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>interface</th>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>computer</th>\n",
        "      <td> 1</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>user</th>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>system</th>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 1</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>response</th>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>time</th>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>eps</th>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>survey</th>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>trees</th>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 1</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>graph</th>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 1</td>\n",
        "      <td> 1</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>minors</th>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 1</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table>\n",
        "</div>"
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       "text": [
        "           0  1  2  3  4  5  6  7  8\n",
        "human      1  0  0  1  0  0  0  0  0\n",
        "interface  1  0  0  0  0  0  0  0  0\n",
        "computer   1  1  0  0  0  0  0  0  0\n",
        "user       0  1  1  0  1  0  0  0  0\n",
        "system     0  1  1  1  0  0  0  0  0\n",
        "response   0  1  0  0  1  0  0  0  0\n",
        "time       0  1  0  0  1  0  0  0  0\n",
        "eps        0  0  1  1  0  0  0  0  0\n",
        "survey     0  1  0  0  0  0  0  0  1\n",
        "trees      0  0  0  0  0  1  1  1  0\n",
        "graph      0  0  0  0  0  0  1  1  1\n",
        "minors     0  0  0  0  0  0  0  1  1"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {
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       "slide_type": "fragment"
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     },
     "source": [
      "Now we can compare the results between ordinary matvec and low-rank matvec."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "query = 'Human computer interaction'\n",
      "qv = np.zeros((nterms))\n",
      "for i in xrange(nterms):\n",
      "    if re.search(rows[i], query, re.IGNORECASE):\n",
      "        qv[i] = 1\n",
      "res1 = qv.dot(term_doc) #Non-compressed search result\n",
      "\n",
      "u, s, v = np.linalg.svd(term_doc)\n",
      "r = 5\n",
      "u = u[:, :r]\n",
      "s = s[:r]\n",
      "v = v[:r, :] #Numpy transposes\n",
      "appr1  = u.dot(np.diag(s)).dot(v)\n",
      "res2 = qv.dot(appr1)\n",
      "res_all = np.vstack((res1, res2)).T #To make two columns\n",
      "print 'There query is:', query, ',the scores are:'\n",
      "pd.DataFrame(res_all, index=docs, columns=['No SVD', 'SVD'])"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "There query is: Human computer interaction ,the scores are:\n"
       ]
      },
      {
       "html": [
        "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
        "<table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>No SVD</th>\n",
        "      <th>SVD</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>Human machine interface for Lab ABC computer applications</th>\n",
        "      <td> 2</td>\n",
        "      <td> 2.087969</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>A survey of user opinions of computer system response time</th>\n",
        "      <td> 1</td>\n",
        "      <td> 0.927427</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>The EPS user interfaces management system</th>\n",
        "      <td> 0</td>\n",
        "      <td> 0.329092</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>System and human system engineering testing of EPS</th>\n",
        "      <td> 1</td>\n",
        "      <td> 0.605656</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>Relation of user-perceived response time on user management</th>\n",
        "      <td> 0</td>\n",
        "      <td>-0.043570</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>The generation of random, binary, unordered trees</th>\n",
        "      <td> 0</td>\n",
        "      <td>-0.008597</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>The intersection graph of paths in trees</th>\n",
        "      <td> 0</td>\n",
        "      <td>-0.027086</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>Graph minors IV: Width of trees and well-quasi-ordering</th>\n",
        "      <td> 0</td>\n",
        "      <td>-0.044239</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>Graph minors: A survey</th>\n",
        "      <td> 0</td>\n",
        "      <td> 0.097481</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table>\n",
        "</div>"
       ],
       "metadata": {},
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       "text": [
        "                                                             No SVD       SVD\n",
        "Human machine interface for Lab ABC computer applications         2  2.087969\n",
        "A survey of user opinions of computer system response time        1  0.927427\n",
        "The EPS user interfaces management system                         0  0.329092\n",
        "System and human system engineering testing of EPS                1  0.605656\n",
        "Relation of user-perceived response time on user management       0 -0.043570\n",
        "The generation of random, binary, unordered trees                 0 -0.008597\n",
        "The intersection graph of paths in trees                          0 -0.027086\n",
        "Graph minors IV: Width of trees and well-quasi-ordering           0 -0.044239\n",
        "Graph minors: A survey                                            0  0.097481"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Application 2: Collaborative filtering\n",
      "\n",
      "Another important (and similar) application comes from **recommender systems**.  \n",
      "Suppose you have a **user-product matrix**: each user puts a rating for a particular product.  \n",
      "The matrix is then **the number of users times the number of products**. The goal is to recommend additional products to be bought for a particular user.  \n",
      "The scheme is the same: we compute the SVD, and the recommendation for each user is just a column of the approximated matrix."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "data_read = np.loadtxt('task2_transact_eval.txt',dtype=np.int32, skiprows=1,delimiter='|')"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Read the data\n",
      "columns = ['SessionNo', 'ItemNo', 'TransType']\n",
      "ds = pd.DataFrame(data=data_read,columns=columns)\n",
      "ds"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
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        "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
        "<table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SessionNo</th>\n",
        "      <th>ItemNo</th>\n",
        "      <th>TransType</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
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        "    <tr>\n",
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        "      <td> 1</td>\n",
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        "      <td>  6498</td>\n",
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        "    <tr>\n",
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        "      <td>      1</td>\n",
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        "      <td>      1</td>\n",
        "      <td> 11396</td>\n",
        "      <td> 2</td>\n",
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        "    <tr>\n",
        "      <th>16    </th>\n",
        "      <td>      3</td>\n",
        "      <td>  5335</td>\n",
        "      <td> 0</td>\n",
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        "    <tr>\n",
        "      <th>17    </th>\n",
        "      <td>      3</td>\n",
        "      <td>   724</td>\n",
        "      <td> 0</td>\n",
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        "    <tr>\n",
        "      <th>18    </th>\n",
        "      <td>      3</td>\n",
        "      <td>  9436</td>\n",
        "      <td> 0</td>\n",
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        "      <td> 0</td>\n",
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        "      <td>      4</td>\n",
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        "      <td> 145685</td>\n",
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        "      <td> 0</td>\n",
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        "    <tr>\n",
        "      <th>985119</th>\n",
        "      <td> 145685</td>\n",
        "      <td> 11800</td>\n",
        "      <td> 0</td>\n",
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        "    <tr>\n",
        "      <th>985120</th>\n",
        "      <td> 145685</td>\n",
        "      <td> 14161</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>985121</th>\n",
        "      <td> 145685</td>\n",
        "      <td>  9266</td>\n",
        "      <td> 0</td>\n",
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        "    <tr>\n",
        "      <th>985122</th>\n",
        "      <td> 145685</td>\n",
        "      <td>  4710</td>\n",
        "      <td> 0</td>\n",
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        "    <tr>\n",
        "      <th>985123</th>\n",
        "      <td> 145685</td>\n",
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        "    <tr>\n",
        "      <th>985124</th>\n",
        "      <td> 145685</td>\n",
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        "      <th>985125</th>\n",
        "      <td> 145685</td>\n",
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        "      <td> 145685</td>\n",
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        "      <td>  7555</td>\n",
        "      <td> 1</td>\n",
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        "      <th>985128</th>\n",
        "      <td> 145685</td>\n",
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        "      <td> 0</td>\n",
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        "      <th>985129</th>\n",
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        "      <td> 1</td>\n",
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        "      <th>985130</th>\n",
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        "      <td> 0</td>\n",
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        "    <tr>\n",
        "      <th>985131</th>\n",
        "      <td> 145687</td>\n",
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        "      <td> 0</td>\n",
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        "    <tr>\n",
        "      <th>985132</th>\n",
        "      <td> 145687</td>\n",
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        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>985133</th>\n",
        "      <td> 145687</td>\n",
        "      <td>  9378</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>985134</th>\n",
        "      <td> 145688</td>\n",
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        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>985135</th>\n",
        "      <td> 145688</td>\n",
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        "      <td> 0</td>\n",
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        "    <tr>\n",
        "      <th>985136</th>\n",
        "      <td> 145688</td>\n",
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        "      <td> 0</td>\n",
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        "    <tr>\n",
        "      <th>985137</th>\n",
        "      <td> 145688</td>\n",
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        "      <td> 0</td>\n",
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        "    <tr>\n",
        "      <th>985138</th>\n",
        "      <td> 145688</td>\n",
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        "      <td> 0</td>\n",
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        "    <tr>\n",
        "      <th>985139</th>\n",
        "      <td> 145688</td>\n",
        "      <td> 13197</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>985140</th>\n",
        "      <td> 145689</td>\n",
        "      <td> 14168</td>\n",
        "      <td> 0</td>\n",
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        "    <tr>\n",
        "      <th>985141</th>\n",
        "      <td> 145689</td>\n",
        "      <td>  5373</td>\n",
        "      <td> 0</td>\n",
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        "    <tr>\n",
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        "      <td> 145690</td>\n",
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        "    <tr>\n",
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        "      <td> 145690</td>\n",
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        "      <td> 0</td>\n",
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        "    <tr>\n",
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        "      <td> 145690</td>\n",
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        "  </tbody>\n",
        "</table>\n",
        "<p>985145 rows \u00d7 3 columns</p>\n",
        "</div>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "        SessionNo  ItemNo  TransType\n",
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        "8               1    3726          1\n",
        "9               1    6498          0\n",
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        "...           ...     ...        ...\n",
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        "\n",
        "[985145 rows x 3 columns]"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ds_buy = ds[ds.TransType==2][['SessionNo', 'ItemNo']]\n",
      "arr = np.array(ds_buy)\n",
      "users, inv_users = np.unique(arr[:, 0], return_inverse=True)\n",
      "products, inv_products = np.unique(arr[:, 1], return_inverse=True)\n",
      "print 'Unique sessions:', len(users), 'Unique products', len(products)\n",
      "#Scipy spars"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Unique sessions: 14164 Unique products 9679\n"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import scipy.sparse\n",
      "import scipy.sparse.linalg\n",
      "%matplotlib inline \n",
      "import matplotlib.pyplot as plt\n",
      "plt.xkcd()\n",
      "nnz = len(inv_users)\n",
      "spmat = scipy.sparse.coo_matrix((np.ones(nnz), (inv_users, inv_products)))\n",
      "r =  2#Approximation rank\n",
      "u, s, v = scipy.sparse.linalg.svds(spmat, r)\n",
      "n = spmat.shape[0]\n",
      "m = spmat.shape[1]\n",
      "q = np.zeros(n)\n",
      "user_id = 5\n",
      "q[user_id] = 1.0\n",
      "\n",
      "qrec = q.T.dot(u).dot(np.diag(s)).dot(v)\n",
      "qrec = qrec / np.max(abs(qrec))\n",
      "qb = spmat.T.dot(q)\n",
      "plt.plot(qrec)\n",
      "plt.plot(qb)\n",
      "plt.xlabel('Product ID')\n",
      "plt.ylabel('Recommendation')"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "<matplotlib.text.Text at 0x1143df050>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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BIlGEKRgM4rvvvksriyxd8vPzccMNN8R9j1dqTMG0gTiIkWUWjOrNFfs1GhpY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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x104436390>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Application 3: Dense matrix compression\n",
      "\n",
      "Dense matrices typically require $N^2$ elements to be stored. For $N \\sim 10^4 - 10^5$ the memory requirements. A low rank approximation can reduces this number of $\\mathcal{O}(Nr)$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "%matplotlib inline\n",
      "import matplotlib.pyplot as plt\n",
      "plt.xkcd()\n",
      "n = 256\n",
      "a = [[1.0/(np.sqrt(i + 3 * j + 0.5)) for i in xrange(n)] for j in xrange(n)]\n",
      "a = np.array(a)\n",
      "u, s, v = np.linalg.svd(a)\n",
      "plt.semilogy(s/s[0])\n",
      "plt.title('Singular values decay for a Hilbert matrix')"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 22,
       "text": [
        "<matplotlib.text.Text at 0x1157e4bd0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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vHJ1OR/v373f4ek9qaqpNE1teXh5NmTKF3NzcxN5C1Rk5cqTYo/NGO3fudOi+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\nHA7Y7XbYbDZuW3w+H4LBYErZzjZv/M+YMQM7d+7kA3xWrFi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       "text": [
        "<matplotlib.figure.Figure at 0x11579fad0>"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## A more realistic example\n",
      "We can try to test on a more realistic matrix, since solving linear systems with Hilbert matrix has little practical sense. Instead, we solve a linear system with a matrix\n",
      "$$A_{ij} = \\frac{1}{i - j + \\frac{1}{2}},$$\n",
      "which corresponds to an integral equation\n",
      "$$\n",
      "    \\int \\frac{q(y)dy}{x - y } = f(x).\n",
      "$$\n",
      "In real life, the equation  \n",
      "$$\n",
      "    \\int_{\\Omega} \\frac{q(y)dy}{\\Vert x - y\\Vert } = f(x),\n",
      "$$\n",
      "is solved, where $\\Omega$ is a surface in 3D. This is used, for example, in modelling integral circuits.\n",
      "Let us see what happens with the singular values."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import numpy as np\n",
      "%matplotlib inline\n",
      "import matplotlib.pyplot as plt\n",
      "plt.xkcd()\n",
      "n = 256\n",
      "a = [[1.0/(i - j + 0.5) for i in xrange(n)] for j in xrange(n)]\n",
      "a = np.array(a)\n",
      "u, s, v = np.linalg.svd(a)\n",
      "plt.plot(s)\n",
      "plt.title('Singular values decay for a Cauchy-Hilbert matrix')\n",
      "plt.tight_layout()"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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+cfnll1NycrJcESUi+uOPP6hZs2Zks9lo1qxZlJubS+Xl5TRp0iQyGAy0bNky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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x115844710>"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "What to do?  \n",
      "The idea is to break the matrix intro blocks  \n",
      "$$\n",
      "    A = \\begin{bmatrix}\n",
      "    A_{11} & A_{12} \\\\\n",
      "    A_{21} & A_{22}\n",
      "    \\end{bmatrix}\n",
      "$$\n",
      "and the blocks $A_{12}$ and $A_{21}$ will be of low-rank!  Let us try that.."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import numpy as np\n",
      "%matplotlib inline\n",
      "import matplotlib.pyplot as plt\n",
      "plt.xkcd()\n",
      "n = 256\n",
      "a = [[1.0/(i - j + 0.5) for i in xrange(n)] for j in xrange(n)]\n",
      "a = np.array(a)\n",
      "\n",
      "a12 = a[:n/2,n/2:]\n",
      "a21 = a[n/2:,:n/2]\n",
      "s12 = np.linalg.svd(a12)[1]\n",
      "s21 = np.linalg.svd(a21)[1]\n",
      "plt.semilogy(s12)\n",
      "plt.semilogy(s21)#Actually, they are the same"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 30,
       "text": [
        "[<matplotlib.lines.Line2D at 0x115c44850>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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BQIC//vWvGAyGDo+KioruFL8dsxmqAlYYOFCdyW337h6vSwgh+lJFRUWnMRHa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\n3w9jx+6fTCgnGVuTGa3fUYv1qZS6e1p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VRVGUBQsWKAMGDFDOOOMM5ZVXXlHmzJmjjBs3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\nQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1156ff350>"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "The whole idea is the applied recursively, leading to $\\mathcal{O}(N \\log N r)$ storage.  \n",
      "A good question is how to compute the approximation without computing all the entries -- here the skeleton decomposition comes into play: to get a rank-$r$ approximation, we can compute $r$ columns and $r$ rows.\n",
      "<img src=\"cross-pic.png\">"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Application 4: Eigenfaces\n",
      "You will have this task in your HW.  The problem setup is similar: you have a database of faces,\n",
      "which you write down as a face-pixel matrix (you can also write this down as a 3D array as well). The goal is given a new face, find a best match to the one in the database. The scheme is the same: you compute the projection onto a low-dimensional subspace using the SVD, and then compare everything in this low-dimensional subspace. The example \"singular vectors\" look similar to the following.\n",
      "<img src=\"face-pc.png\" width=50%>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Higher dimensions\n",
      "There are several ways how to generalize SVD to multidimensional arrays (tensors). In data mining, this is very natural. For example, in real recommender system, the data are organized as  \n",
      "\n",
      "**user-product-action**  \n",
      "\n",
      "where action is either \"watch\" or \"put into the bucket\" or \"buy\".  \n",
      "It is natural to organize such data into a 3D tensor of size  \n",
      "\n",
      "\n",
      "**number of users $\\times$ number of products $\\times$ number of actions.**\n",
      "\n",
      "This is what High Order SVD (HOSVD) is used.  \n",
      "The idea is to decompose a tensor in the form  \n",
      "$$\n",
      "  A_{ijk} \\approx \\sum_{\\alpha \\beta \\gamma} U_{i \\alpha} V_{j \\beta} W_{k \\gamma} g_{\\alpha \\beta \\gamma}.\n",
      "$$\n",
      "This can be done by considering three **unfoldings** of a tensor and computing its left singular matrices. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "##### Questions?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.core.display import HTML\n",
      "def css_styling():\n",
      "    styles = open(\"./styles/custom.css\", \"r\").read()\n",
      "    return HTML(styles)\n",
      "css_styling()"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "outputs": [
      {
       "html": [
        "<link href='http://fonts.googleapis.com/css?family=Fenix' rel='stylesheet' type='text/css'>\n",
        "<link href='http://fonts.googleapis.com/css?family=Alegreya+Sans:100,300,400,500,700,800,900,100italic,300italic,400italic,500italic,700italic,800italic,900italic' rel='stylesheet' type='text/css'>\n",
        "<link href='http://fonts.googleapis.com/css?family=Source+Code+Pro:300,400' rel='stylesheet' type='text/css'>\n",
        "<style>\n",
        "    @font-face {\n",
        "        font-family: \"Computer Modern\";\n",
        "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
        "    }\n",
        "    div.cell{\n",
        "        /*width:80%;*/\n",
        "        /*margin-left:auto !important;\n",
        "        margin-right:auto;*/\n",
        "    }\n",
        "    h1 {\n",
        "        font-family: 'Alegreya Sans', sans-serif;\n",
        "    }\n",
        "    h2 {\n",
        "        font-family: 'Fenix', serif;\n",
        "    }\n",
        "    h3{\n",
        "\t\tfont-family: 'Fenix', serif;\n",
        "        margin-top:12px;\n",
        "        margin-bottom: 3px;\n",
        "       }\n",
        "\th4{\n",
        "\t\tfont-family: 'Fenix', serif;\n",
        "       }\n",
        "    h5 {\n",
        "        font-family: 'Alegreya Sans', sans-serif;\n",
        "    }\t   \n",
        "    div.text_cell_render{\n",
        "        font-family: 'Alegreya Sans',Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
        "        line-height: 1.2;\n",
        "        font-size: 160%;\n",
        "        /*width:70%;*/\n",
        "        /*margin-left:auto;*/\n",
        "        margin-right:auto;\n",
        "    }\n",
        "    .CodeMirror{\n",
        "            font-family: \"Source Code Pro\";\n",
        "\t\t\tfont-size: 90%;\n",
        "    }\n",
        "/*    .prompt{\n",
        "        display: None;\n",
        "    }*/\n",
        "    .text_cell_render h1 {\n",
        "        font-weight: 200;\n",
        "        font-size: 50pt;\n",
        "\t\tline-height: 110%;\n",
        "        color:#CD2305;\n",
        "        margin-bottom: 0.5em;\n",
        "        margin-top: 0.5em;\n",
        "        display: block;\n",
        "    }\t\n",
        "    .text_cell_render h5 {\n",
        "        font-weight: 300;\n",
        "        font-size: 16pt;\n",
        "        color: #CD2305;\n",
        "        font-style: italic;\n",
        "        margin-bottom: .5em;\n",
        "        margin-top: 0.5em;\n",
        "        display: block;\n",
        "    }\n",
        "    \n",
        "    .warning{\n",
        "        color: rgb( 240, 20, 20 )\n",
        "        }  \n",
        "</style>\n",
        "<script>\n",
        "    MathJax.Hub.Config({\n",
        "                        TeX: {\n",
        "                           extensions: [\"AMSmath.js\"]\n",
        "                           },\n",
        "                tex2jax: {\n",
        "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
        "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
        "                },\n",
        "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
        "                \"HTML-CSS\": {\n",
        "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
        "                }\n",
        "        });\n",
        "</script>\n"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 137,
       "text": [
        "<IPython.core.display.HTML at 0x1224ad390>"
       ]
      }
     ],
     "prompt_number": 137
    }
   ],
   "metadata": {}
  }
 ]
}