Properties of blocking eigen decomposition

blocking_eigenvecs

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  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import pylab\n",
      "import numpy as np\n",
      "import scipy as sp\n",
      "\n",
      "from IPython.display import display, Math, Latex\n",
      "\n",
      "#My matrix is given by:\n",
      "\n",
      "display(Latex(r\"\"\"\n",
      "Basic reminder for working with subsets of eigen-decompositions:\n",
      "\n",
      "$$\n",
      "U = \\left[\n",
      "\\begin{array}{c}\n",
      "U_{1} \\\\\n",
      "U_{2} \n",
      "\\end{array}\\right]\n",
      "$$\n",
      "\n",
      "$$\n",
      "U^T U = \\left[\n",
      "\\begin{array}{c}\n",
      "U_{1} \\\\\n",
      "U_{2} \n",
      "\\end{array}\\right]^T \n",
      "\\left[\n",
      "\\begin{array}{c}\n",
      "U_{1} \\\\\n",
      "U_{2} \n",
      "\\end{array}\\right] =\n",
      "U_1 U_1^T + U_2 U_2^T = I \\Rightarrow U_1 U_1^T = I - U_2 U_2^T\n",
      "$$\n",
      "\n",
      "$$\n",
      "U^T U = \\left[\n",
      "\\begin{array}{c}\n",
      "U_{1} \\\\\n",
      "U_{2} \n",
      "\\end{array}\\right]\n",
      "\\left[\n",
      "\\begin{array}{c}\n",
      "U_{1} \\\\\n",
      "U_{2} \n",
      "\\end{array}\\right]^T = \n",
      "\\left[\n",
      "\\begin{array}{cc}\n",
      "\\overbrace{U_1 U_1^T}^{I} & \\overbrace{U_1 U_2^T}^{=0 \\text{ orthogonality}}\\\\\n",
      "U_2 U_1^T & U_2 U_2^t \n",
      "\\end{array}\\right]= \n",
      "\\left[\n",
      "\\begin{array}{cc}\n",
      "I & 0 \\\\\n",
      "0 & I\n",
      "\\end{array}\\right]\n",
      "$$\n",
      "\n",
      "\"\"\"))\n",
      "\n",
      "\n",
      "\n",
      "#display(Math(r'F(k) = \\int_{-\\infty}^{\\infty} f(x) e^{2\\pi i k} dx'))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "\n",
        "Basic reminder for working with subsets of eigen-decompositions:\n",
        "\n",
        "$$\n",
        "U = \\left[\n",
        "\\begin{array}{c}\n",
        "U_{1} \\\\\n",
        "U_{2} \n",
        "\\end{array}\\right]\n",
        "$$\n",
        "\n",
        "$$\n",
        "U^T U = \\left[\n",
        "\\begin{array}{c}\n",
        "U_{1} \\\\\n",
        "U_{2} \n",
        "\\end{array}\\right]^T \n",
        "\\left[\n",
        "\\begin{array}{c}\n",
        "U_{1} \\\\\n",
        "U_{2} \n",
        "\\end{array}\\right] =\n",
        "U_1 U_1^T + U_2 U_2^T = I \\Rightarrow U_1 U_1^T = I - U_2 U_2^T\n",
        "$$\n",
        "\n",
        "$$\n",
        "U^T U = \\left[\n",
        "\\begin{array}{c}\n",
        "U_{1} \\\\\n",
        "U_{2} \n",
        "\\end{array}\\right]\n",
        "\\left[\n",
        "\\begin{array}{c}\n",
        "U_{1} \\\\\n",
        "U_{2} \n",
        "\\end{array}\\right]^T = \n",
        "\\left[\n",
        "\\begin{array}{cc}\n",
        "\\overbrace{U_1 U_1^T}^{I} & \\overbrace{U_1 U_2^T}^{=0 \\text{ orthogonality}}\\\\\n",
        "U_2 U_1^T & U_2 U_2^t \n",
        "\\end{array}\\right]= \n",
        "\\left[\n",
        "\\begin{array}{cc}\n",
        "I & 0 \\\\\n",
        "0 & I\n",
        "\\end{array}\\right]\n",
        "$$\n",
        "\n"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x9c58e10>"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# eigenvectors are orthogonal\n",
      "\n",
      "def load_data(num_ind):\n",
      "    \n",
      "    X = np.random.random_sample((num_ind, 1000))\n",
      "    y = np.random.random_sample(num_ind)\n",
      "\n",
      "    X -= X.mean(axis=0)\n",
      "    y -= y.mean()\n",
      "\n",
      "    return X, y\n",
      "\n",
      "\n",
      "num_ind = 10\n",
      "X, y = load_data(num_ind)\n",
      "\n",
      "K = X.dot(X.T)\n",
      "\n",
      "#TODO: establish relationship\n",
      "U0, S0, V0 = np.linalg.svd(X)\n",
      "\n",
      "S, U = np.linalg.eigh(K)\n",
      "\n",
      "print \n",
      "\n",
      "idx = np.random.permutation(np.arange(num_ind))[0:num_ind/2]\n",
      "\n",
      "UUT = U.dot(U.T)\n",
      "UTU = U.T.dot(U)\n",
      "\n",
      "pylab.imshow(UUT, interpolation=\"nearest\")\n",
      "pylab.title(r\"UU^T\")\n",
      "pylab.show()\n",
      "\n",
      "\n",
      "pylab.imshow(UTU, interpolation=\"nearest\")\n",
      "pylab.title(r\"U^TU\")\n",
      "pylab.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
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       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPYAAAEKCAYAAAAhNageAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x9ad6470>"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# we show that a subset only maintains identity in one direction\n",
      "\n",
      "idx = np.random.permutation(np.arange(num_ind))[0:num_ind/2]\n",
      "U1 = U[~idx,:]\n",
      "U2 = U[idx,:]\n",
      "\n",
      "UUT = U2.dot(U2.T)\n",
      "UTU = U2.T.dot(U2)\n",
      "\n",
      "pylab.imshow(UUT, interpolation=\"nearest\")\n",
      "pylab.title(r\"U2U2^T\")\n",
      "pylab.show()\n",
      "\n",
      "\n",
      "pylab.imshow(UTU, interpolation=\"nearest\")\n",
      "pylab.title(r\"U2^TU2\")\n",
      "pylab.show()\n",
      "\n",
      "pylab.imshow(UUT + U1.dot(U1.T), interpolation=\"nearest\")\n",
      "pylab.title(r\"U1U1^T + U2U2^T\")\n",
      "pylab.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPYAAAEKCAYAAAAhNageAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x9dbae80>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x9af2128>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x747f048>"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}