{"nbformat": 3, "worksheets": [{"cells": [{"cell_type": "markdown", "metadata": {}, "source": ["# Finding the SVD"]}, {"cell_type": "code", "outputs": [], "language": "python", "prompt_number": 10, "input": ["import numpy as np\n", "import numpy.linalg as la\n", "import matplotlib.pyplot as pt"], "metadata": {}, "collapsed": false}, {"cell_type": "code", "outputs": [], "language": "python", "prompt_number": 11, "input": ["np.random.seed(27)\n", "X = np.random.randn(2,2)"], "metadata": {}, "collapsed": false}, {"cell_type": "code", "outputs": [], "language": "python", "prompt_number": 7, "input": ["phi = np.linspace(0, 2*np.pi, 100)\n", "norm1_vecs = np.array([\n", "    np.cos(phi),\n", "    np.sin(phi)\n", "])"], "metadata": {}, "collapsed": false}, {"cell_type": "code", "outputs": [{"text": ["<matplotlib.figure.Figure at 0x7f10a2d09080>"], "metadata": {}, "png": 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bDH63bhAtUe1kMan28ddU1TXR8zVAWUlbnwLuBIri6CtmnnwSbrzRjb6TuSxf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\nnCzDDb/jOE6W4YbfcRwny/j/0cdlY8slunIAAAAASUVORK5CYII=\n", "output_type": "display_data"}], "language": "python", "prompt_number": 8, "input": ["X_norm1_vecs = X.dot(norm1_vecs)\n", "\n", "pt.grid()\n", "pt.plot(X_norm1_vecs[0], X_norm1_vecs[1])\n", "pt.gca().set_aspect(\"equal\")"], "metadata": {}, "collapsed": false}, {"cell_type": "markdown", "metadata": {}, "source": ["Approximately, what is $\\|X\\|_2$?"]}, {"cell_type": "code", "outputs": [{"text": ["1.4317821063276355"], "metadata": {}, "prompt_number": 15, "output_type": "pyout"}], "language": "python", "prompt_number": 15, "input": ["np.sqrt(1.3**2 + 0.6**2)"], "metadata": {}, "collapsed": false}, {"cell_type": "code", "outputs": [{"text": ["1.4309926780552626"], "metadata": {}, "prompt_number": 16, "output_type": "pyout"}], "language": "python", "prompt_number": 16, "input": ["la.norm(X, 2)"], "metadata": {}, "collapsed": false}, {"cell_type": "markdown", "metadata": {}, "source": ["--------------\n", "Let's find the vector $v_1$ that achieves that.\n", "\n", "First, find the norms of all the $Xn$, where $n$ are the norm-1 vectors we generated above:"]}, {"cell_type": "code", "outputs": [{"text": ["1.4308557961109123"], "metadata": {}, "prompt_number": 18, "output_type": "pyout"}], "language": "python", "prompt_number": 18, "input": ["norm_X_norm1_vecs = np.sqrt(np.sum(X_norm1_vecs**2, axis=0))\n"], "metadata": {}, "collapsed": false}, {"cell_type": "markdown", "metadata": {}, "source": ["Next, find the approximate 2-norm of X from this as `approx_norm`:"]}, {"cell_type": "code", "outputs": [{"text": ["1.4308557961109123"], "metadata": {}, "prompt_number": 19, "output_type": "pyout"}], "language": "python", "prompt_number": 19, "input": ["approx_norm = np.max(norm_X_norm1_vecs)\n", "approx_norm"], "metadata": {}, "collapsed": false}, {"cell_type": "markdown", "metadata": {}, "source": ["And find the vector $v_1$ that achieves `approx_norm`:"]}, {"cell_type": "code", "outputs": [{"text": ["(array([48]),)"], "metadata": {}, "prompt_number": 22, "output_type": "pyout"}], "language": "python", "prompt_number": 22, "input": ["np.where(norm_X_norm1_vecs == approx_norm)"], "metadata": {}, "collapsed": false}, {"cell_type": "code", "outputs": [], "language": "python", "prompt_number": 25, "input": ["v1 = norm1_vecs[:, 48]"], "metadata": {}, "collapsed": false}, {"cell_type": "code", "outputs": [{"text": ["0.99999999999999989"], "metadata": {}, "prompt_number": 28, "output_type": "pyout"}], "language": "python", "prompt_number": 28, "input": ["la.norm(v1, 2)"], "metadata": {}, "collapsed": false}, {"cell_type": "code", "outputs": [{"text": ["<matplotlib.figure.Figure at 0x7f10a2cc3d68>"], "metadata": {}, "png": 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"output_type": "display_data"}], "language": "python", "prompt_number": 27, "input": ["\n", "X_v1 = X.dot(v1)\n", "\n", "pt.grid()\n", "pt.plot(X_norm1_vecs[0], X_norm1_vecs[1])\n", "pt.arrow(0, 0, X_v1[0], X_v1[1])\n", "pt.gca().set_aspect(\"equal\")"], "metadata": {}, "collapsed": false}, {"cell_type": "markdown", "metadata": {}, "source": ["------------------\n", "Now $v_2$ has to be orthogonal--that doesn't leave much choice in 2D:"]}, {"cell_type": "code", "outputs": [], "language": "python", "prompt_number": 32, "input": ["v2 = np.array([v1[1], -v1[0]])"], "metadata": {}, "collapsed": false}, {"cell_type": "code", "outputs": [{"text": ["1.0\n", "0.0\n"], "stream": "stdout", "output_type": "stream"}], "language": "python", "prompt_number": 33, "input": ["print(la.norm(v2, 2))\n", "print(v1.dot(v2))"], "metadata": {}, "collapsed": false}, {"cell_type": "markdown", "metadata": {}, "source": ["------------\n", "Now define the matrices $V$ and $\\Sigma$:"]}, {"cell_type": "code", "outputs": [{"text": ["[[-0.99547192  0.09505604]\n", " [ 0.09505604  0.99547192]]\n", "[[ 1.4308558   0.        ]\n", " [ 0.          0.48749067]]\n"], "stream": "stdout", "output_type": "stream"}], "language": "python", "prompt_number": 36, "input": ["V = np.array([v1, v2]).T\n", "print(V)\n", "\n", "Sigma = np.diag([\n", "    la.norm(X.dot(v1)),\n", "    la.norm(X.dot(v2))\n", "])\n", "print(Sigma)"], "metadata": {}, "collapsed": false}, {"cell_type": "code", "outputs": [], "language": "python", "prompt_number": 45, "input": ["U = X.dot(V).dot(la.inv(Sigma))"], "metadata": {}, "collapsed": false}, {"cell_type": "markdown", "metadata": {}, "source": ["Now watch this:"]}, {"cell_type": "code", "outputs": [{"text": ["array([[ 0.97276264,  0.02674873],\n", "       [ 0.02674873,  1.02723736]])"], "metadata": {}, "prompt_number": 48, "output_type": "pyout"}], "language": "python", "prompt_number": 48, "input": ["U.dot(U.T)"], "metadata": {}, "collapsed": false}, {"cell_type": "markdown", "metadata": {}, "source": ["Notice anything? (It's not very accurate, but still visible.)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["--------------\n", "What we have discovered is called the *singular value decomposition (\"SVD\")*."]}, {"cell_type": "code", "outputs": [{"text": ["(array([[-0.91254817, -0.40896923],\n", "        [-0.40896923,  0.91254817]]),\n", " array([ 1.43099268,  0.48708872]),\n", " array([[-0.99676245,  0.08040283],\n", "        [ 0.08040283,  0.99676245]]))"], "metadata": {}, "prompt_number": 47, "output_type": "pyout"}], "language": "python", "prompt_number": 47, "input": ["la.svd(X)"], "metadata": {}, "collapsed": false}, {"cell_type": "code", "outputs": [{"text": ["[[-0.9145846  -0.36918512]\n", " [-0.40439462  0.92935588]]\n", "[[ 1.4308558   0.        ]\n", " [ 0.          0.48749067]]\n", "[[-0.99547192  0.09505604]\n", " [ 0.09505604  0.99547192]]\n"], "stream": "stdout", "output_type": "stream"}], "language": "python", "prompt_number": 53, "input": ["print(U)\n", "print(Sigma)\n", "print(V)"], "metadata": {}, "collapsed": false}], "metadata": {}}], "metadata": {"name": "", "signature": "sha256:7c90e0de6146bb41bedebb25735e6f0bb71860df65e6312e232b2dd7e4c48ba3"}, "nbformat_minor": 0}