{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Rank-1 Approximation"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import numpy.linalg as la\n",
        "import matplotlib.pyplot as pt"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 67,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "np.random.seed(17)\n",
        "n = 10\n",
        "X = np.random.randn(2, n)\n",
        "\n",
        "X[1] = 0.7*X[0] + 0.2 * X[1]\n",
        "\n",
        "# uncomment this for a different data set\n",
        "#X[0] = 0.1 * X[0]"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 47,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f0e4bcf1dd8>]"
            ]
          },
          "execution_count": 47,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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UhtfqkaSKea0eSdJADv6Gauj9asgI52fOLq95fz7uZ5dqydmG1+OXCvGa96qFHb9UiNe8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\nTkSuQmCC\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f0e4bcfccf8>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "pt.figure(figsize=(6,6))\n",
        "pt.gca().set_aspect(\"equal\")\n",
        "pt.xlim([-2, 2])\n",
        "pt.ylim([-2, 2])\n",
        "pt.grid()\n",
        "pt.plot(X[0], X[1], \"o\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Now compute the SVD. Use `numpy.linalg.svd(..., full_matrices=False)`."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 52,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "U, sigma, VT = la.svd(X, full_matrices=False)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Now find the vectors `u` and `v`:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 53,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "u = U[:, 0]\n",
        "v = VT[0]"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Now find `X1`:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 61,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "X1 = np.outer(u, v) * sigma[0]"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 66,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f0e4ba4a2e8>"
            ]
          },
          "execution_count": 66,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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OYqjwi4hUGPX4RUQCph6/iIjkpMJfoBD6fiFkBOWMmnJGK5ScxVDhFxGpMOrx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\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f0e4b9aa898>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "\n",
        "pt.figure(figsize=(6,6))\n",
        "\n",
        "pt.arrow(0, 0, u[0], u[1], lw=3)\n",
        "\n",
        "pt.gca().set_aspect(\"equal\")\n",
        "pt.xlim([-2, 2])\n",
        "pt.ylim([-2, 2])\n",
        "pt.grid()\n",
        "\n",
        "pt.plot(X[0], X[1], \"ob\", label=\"X\")\n",
        "pt.plot(X1[0], X1[1], \"og\", label=\"X1\")\n",
        "pt.legend(loc=\"best\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "* Is this the same as least-squares data fitting?\n",
        "* Would least-squares data fitting deal with the second data set above?"
      ]
    }
  ],
  "metadata": {},
  "nbformat": 4,
  "nbformat_minor": 0
}