tobydriscoll/TB-Lecture-04.ipynb

## TB-Lecture-04.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lecture 4: Singular value decomposition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Geometric interpretation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The SVD has many possible interpretations and uses. We can visualize the geometry of the SVD using the same 2D picture used to illustrate the induced 2-norm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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2j1Iw0jqHuKB2H4WePZ1BxI+CsmTXQ20kbyi7h19tsQ6RDKWlxGLWIcSYP9br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\nEREnqJBERMQJKiQREXHC/wefhAgrwLN8DwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = [-2 0; -1 3];\n",
    "t = linspace(0,2*pi,300);\n",
    "x1 = cos(t);  x2 = sin(t);\n",
    "x = [x1;x2];  Ax = A*x;\n",
    "subplot(121), scatter(x1,x2,4,t,'.'), axis equal, axis off, title('x')\n",
    "subplot(122), scatter(Ax(1,:),Ax(2,:),4,t,'.'), axis equal, axis off, title('Ax')\n",
    "colormap(hsv)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **left singular vectors** and their associated **singular values** are found from the major and minor axes of the ellipse. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sigma =\n",
      "\n",
      "    3.2566    1.8424\n",
      "\n",
      "\n",
      "U =\n",
      "\n",
      "    0.2883    0.9558\n",
      "    0.9576   -0.2939\n"
     ]
    }
   ],
   "source": [
    "normAx = sqrt( sum(Ax.^2,1) );  % 2-norms of all the vectors\n",
    "[sigma1,k1] = max(normAx);  [sigma2,k2] = min(normAx);\n",
    "sigma = [ sigma1 sigma2 ]\n",
    "U = [ Ax(:,k1)/sigma1 Ax(:,k2)/sigma2 ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **right singular vectors** are the pre-images of those vectors on the ellipse."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "V =\n",
      "\n",
      "   -0.4694   -0.8805\n",
      "    0.8830   -0.4740\n"
     ]
    }
   ],
   "source": [
    "V = [ x(:,k1) x(:,k2) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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cIQFYgPgduEN0Ch+xIAVKTEzM/Pnzd+/e3dTUBKCystJms91zzz2icxEFkq0RXVXDkDIM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\nRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "subplot(121), scatter(x1,x2,4,t,'.'), axis equal, axis off, title('x')\n",
    "hold on, plot([0 0; V(1,:)],[0 0; V(2,:)], 'k' )\n",
    "text(1.2*V(1,:),1.2*V(2,:),{'v_1','v_2'})\n",
    "subplot(122), scatter(Ax(1,:),Ax(2,:),4,t,'.'), axis equal, axis off, title('Ax')\n",
    "hold on, plot([0 0; U(1,:).*sigma],[0 0; U(2,:).*sigma], 'k' )\n",
    "text(1.1*U(1,:).*sigma,1.1*U(2,:).*sigma,{'\\sigma_1 u_1','\\sigma_2 u_2'})\n",
    "colormap(hsv)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Having searched over only a finite number of vectors, we have approximately (up to choices of signs in the singular vectors) found the SVD of this matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "U =\n",
      "\n",
      "    0.2898    0.9571\n",
      "    0.9571   -0.2898\n",
      "\n",
      "\n",
      "S =\n",
      "\n",
      "    3.2566         0\n",
      "         0    1.8424\n",
      "\n",
      "\n",
      "V =\n",
      "\n",
      "   -0.4719   -0.8817\n",
      "    0.8817   -0.4719\n"
     ]
    }
   ],
   "source": [
    "[U,S,V] = svd(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Full versus thin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the case of a rectangular matrix, some of the SVD is essentially dead weight. We consider (as always in this text) the case with $m>n$, a tall and skinny matrix. The range of $A$ can have at most only $n$ dimensions. The last $m-n$ columns of $U$ therefore describe the remainder of $\\mathbb{C}^m$ in which the range is embedded (i.e., the orthogonal complement). That description can be changed without affecting the matrix at all.\n",
    "\n",
    "Algebraically, we have \n",
    "\n",
    "$$U = \\bigl[ U_1 \\: U_2 \\bigr], \\qquad S = \\begin{bmatrix} S_1 \\\\ 0 \\end{bmatrix},$$\n",
    "\n",
    "where the breaks occur after $n$ columns in $U$ and $n$ rows in $S$. Therefore $US=U_1S_1$, and we change nothing by replacing the SVD with $U_1S_1V^*$. The book calls this the **reduced SVD**, though it is also called the **thin SVD**. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "U =\n",
      "\n",
      "   -0.4360   -0.2481   -0.6419    0.3068   -0.4848    0.0844\n",
      "   -0.4139   -0.3658    0.0391   -0.7713   -0.0349   -0.3120\n",
      "   -0.4254    0.5159    0.0610    0.2613    0.0834   -0.6884\n",
      "   -0.4385   -0.3189    0.7227    0.3450   -0.1445    0.2094\n",
      "   -0.2543   -0.3155   -0.2428    0.2029    0.8540    0.0798\n",
      "   -0.4480    0.5807   -0.0388   -0.2872    0.0814    0.6095\n",
      "\n",
      "\n",
      "S =\n",
      "\n",
      "    2.6512         0         0\n",
      "         0    1.0597         0\n",
      "         0         0    0.7824\n",
      "         0         0         0\n",
      "         0         0         0\n",
      "         0         0         0\n",
      "\n",
      "\n",
      "V =\n",
      "\n",
      "   -0.5240   -0.8512    0.0293\n",
      "   -0.6235    0.4068    0.6677\n",
      "   -0.5802    0.3316   -0.7439\n",
      "\n",
      "\n",
      "U1 =\n",
      "\n",
      "   -0.4360   -0.2481   -0.6419\n",
      "   -0.4139   -0.3658    0.0391\n",
      "   -0.4254    0.5159    0.0610\n",
      "   -0.4385   -0.3189    0.7227\n",
      "   -0.2543   -0.3155   -0.2428\n",
      "   -0.4480    0.5807   -0.0388\n",
      "\n",
      "\n",
      "S1 =\n",
      "\n",
      "    2.6512         0         0\n",
      "         0    1.0597         0\n",
      "         0         0    0.7824\n",
      "\n",
      "\n",
      "V =\n",
      "\n",
      "   -0.5240   -0.8512    0.0293\n",
      "   -0.6235    0.4068    0.6677\n",
      "   -0.5802    0.3316   -0.7439\n"
     ]
    }
   ],
   "source": [
    "A = rand(6,3); \n",
    "[U,S,V] = svd(A)       % full SVD\n",
    "[U1,S1,V] = svd(A,0)   % thin SVD"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The price we pay is that while the columns of $U_1$ are orthonormal, we can't call it a unitary matrix because it isn't square. Take note:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "    1.0000    0.0000    0.0000\n",
      "    0.0000    1.0000   -0.0000\n",
      "    0.0000   -0.0000    1.0000\n",
      "\n",
      "\n",
      "ans =\n",
      "\n",
      "    0.6637    0.2461    0.0184   -0.1936    0.3450    0.0761\n",
      "    0.2461    0.3066   -0.0103    0.3264    0.2112   -0.0286\n",
      "    0.0184   -0.0103    0.4509    0.0661   -0.0694    0.4878\n",
      "   -0.1936    0.3264    0.0661    0.8162    0.0367   -0.0168\n",
      "    0.3450    0.2112   -0.0694    0.0367    0.2232   -0.0599\n",
      "    0.0761   -0.0286    0.4878   -0.0168   -0.0599    0.5394\n"
     ]
    }
   ],
   "source": [
    "U1'*U1    % n by n identity\n",
    "U1*U1'    % NOT the m by m identity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll discuss the significance of $U_1U_1^*$ in due course."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Matlab",
   "language": "matlab",
   "name": "matlab"
  },
  "language_info": {
   "codemirror_mode": "octave",
   "file_extension": ".m",
   "help_links": [
    {
     "text": "MetaKernel Magics",
     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
    }
   ],
   "mimetype": "text/x-octave",
   "name": "matlab",
   "version": "0.11.0"
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