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    "Douglas Rachford Proximal Splitting\n",
    "===================================\n",
    "\n",
    "*Important:* Please read the [installation page](http://gpeyre.github.io/numerical-tours/installation_matlab/) for details about how to install the toolboxes.\n",
    "$\\newcommand{\\dotp}[2]{\\langle #1, #2 \\rangle}$\n",
    "$\\newcommand{\\enscond}[2]{\\lbrace #1, #2 \\rbrace}$\n",
    "$\\newcommand{\\pd}[2]{ \\frac{ \\partial #1}{\\partial #2} }$\n",
    "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n",
    "$\\newcommand{\\umax}[1]{\\underset{#1}{\\max}\\;}$\n",
    "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n",
    "$\\newcommand{\\uargmin}[1]{\\underset{#1}{argmin}\\;}$\n",
    "$\\newcommand{\\norm}[1]{\\|#1\\|}$\n",
    "$\\newcommand{\\abs}[1]{\\left|#1\\right|}$\n",
    "$\\newcommand{\\choice}[1]{ \\left\\{  \\begin{array}{l} #1 \\end{array} \\right. }$\n",
    "$\\newcommand{\\pa}[1]{\\left(#1\\right)}$\n",
    "$\\newcommand{\\diag}[1]{{diag}\\left( #1 \\right)}$\n",
    "$\\newcommand{\\qandq}{\\quad\\text{and}\\quad}$\n",
    "$\\newcommand{\\qwhereq}{\\quad\\text{where}\\quad}$\n",
    "$\\newcommand{\\qifq}{ \\quad \\text{if} \\quad }$\n",
    "$\\newcommand{\\qarrq}{ \\quad \\Longrightarrow \\quad }$\n",
    "$\\newcommand{\\ZZ}{\\mathbb{Z}}$\n",
    "$\\newcommand{\\CC}{\\mathbb{C}}$\n",
    "$\\newcommand{\\RR}{\\mathbb{R}}$\n",
    "$\\newcommand{\\EE}{\\mathbb{E}}$\n",
    "$\\newcommand{\\Zz}{\\mathcal{Z}}$\n",
    "$\\newcommand{\\Ww}{\\mathcal{W}}$\n",
    "$\\newcommand{\\Vv}{\\mathcal{V}}$\n",
    "$\\newcommand{\\Nn}{\\mathcal{N}}$\n",
    "$\\newcommand{\\NN}{\\mathcal{N}}$\n",
    "$\\newcommand{\\Hh}{\\mathcal{H}}$\n",
    "$\\newcommand{\\Bb}{\\mathcal{B}}$\n",
    "$\\newcommand{\\Ee}{\\mathcal{E}}$\n",
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    "$\\newcommand{\\Gg}{\\mathcal{G}}$\n",
    "$\\newcommand{\\Ss}{\\mathcal{S}}$\n",
    "$\\newcommand{\\Pp}{\\mathcal{P}}$\n",
    "$\\newcommand{\\Ff}{\\mathcal{F}}$\n",
    "$\\newcommand{\\Xx}{\\mathcal{X}}$\n",
    "$\\newcommand{\\Mm}{\\mathcal{M}}$\n",
    "$\\newcommand{\\Ii}{\\mathcal{I}}$\n",
    "$\\newcommand{\\Dd}{\\mathcal{D}}$\n",
    "$\\newcommand{\\Ll}{\\mathcal{L}}$\n",
    "$\\newcommand{\\Tt}{\\mathcal{T}}$\n",
    "$\\newcommand{\\si}{\\sigma}$\n",
    "$\\newcommand{\\al}{\\alpha}$\n",
    "$\\newcommand{\\la}{\\lambda}$\n",
    "$\\newcommand{\\ga}{\\gamma}$\n",
    "$\\newcommand{\\Ga}{\\Gamma}$\n",
    "$\\newcommand{\\La}{\\Lambda}$\n",
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    "$\\newcommand{\\om}{\\omega}$\n",
    "$\\newcommand{\\Om}{\\Omega}$\n"
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   "source": [
    "This numerical tour presents the Douglas-Rachford (DR) algorithm to\n",
    "minimize the sum of two simple functions. It shows an\n",
    "application to\n",
    "reconstruction of exactly sparse signal from noiseless measurement using\n",
    "$\\ell^1$ minimization."
   ]
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   "execution_count": 1,
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   "source": [
    "library(pracma)\n",
    "\n",
    "options(warn=-1) # turns off warnings, to turn on: \"options(warn=0)\"\n",
    "\n",
    "# Importing the libraries\n",
    "for (f in list.files(path=\"nt_toolbox/toolbox_general/\", pattern=\"*.R\")) {\n",
    "    source(paste(\"nt_toolbox/toolbox_general/\", f, sep=\"\"))\n",
    "}\n",
    "for (f in list.files(path=\"nt_toolbox/toolbox_signal/\", pattern=\"*.R\")) {\n",
    "    source(paste(\"nt_toolbox/toolbox_signal/\", f, sep=\"\"))\n",
    "}\n",
    "for (f in list.files(path=\"nt_toolbox/toolbox_graph/\", pattern=\"*.R\")) {\n",
    "    source(paste(\"nt_toolbox/toolbox_graph/\", f, sep=\"\"))\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Douglas-Rachford Algorithm\n",
    "--------------------------\n",
    "The Douglas-Rachford (DR) algorithm is an iterative scheme to minimize\n",
    "functionals of the form\n",
    "$$ \\umin{x} F(x) + G(x) $$\n",
    "where $F$ and $G$ are convex functions for which one is able to\n",
    "comptue the proximal mappings $ \\text{prox}_{\\gamma F} $ and\n",
    "$ \\text{prox}_{\\gamma G} $ which are defined as\n",
    "$$ \\text{prox}_{\\gamma F}(x) = \\text{argmin}_{y} \\frac{1}{2}\\norm{x-y}^2 + \\ga F(y) $$\n",
    "(the same definition applies also for $G$).\n",
    "\n",
    "\n",
    "The important point is that $F$ and $G$ do not need to be smooth.\n",
    "One only needs them to be \"proximable\".\n",
    "\n",
    "\n",
    "This algorithm was introduced in:\n",
    "\n",
    "\n",
    "P. L. Lions and B. Mercier\n",
    "_Splitting Algorithms for the Sum of Two Nonlinear Operators_\n",
    "SIAM Journal on Numerical Analysis\n",
    "Vol. 16, No. 6 (Dec., 1979), pp. 964-979\n",
    "\n",
    "\n",
    "as a generalization of an algorithm introduced by Douglas and Rachford in\n",
    "the case of quadratic minimization (which corresponds to the solution of\n",
    "a linear system).\n",
    "\n",
    "\n",
    "To learn more about this algorithm, you can read:\n",
    "\n",
    "\n",
    "Patrick L. Combettes and Jean-Christophe Pesquet,\n",
    "_Proximal Splitting Methods in Signal Processing_,\n",
    "in: Fixed-Point Algorithms for Inverse\n",
    "Problems in Science and Engineering, New York: Springer-Verlag, 2010.\n",
    "\n",
    "\n",
    "\n",
    "A DR iteration reads\n",
    "$$ \\tilde x_{k+1} = \\pa{1-\\frac{\\mu}{2}} \\tilde x_k +\n",
    "  \\frac{\\mu}{2} \\text{rprox}_{\\gamma G}( \\text{rprox}_{\\gamma F}(\\tilde x_k)  )\n",
    "  \\qandq x_{k+1} = \\text{prox}_{\\gamma F}(\\tilde x_{k+1},) $$\n",
    "\n",
    "\n",
    "\n",
    "We have used the following shortcuts:\n",
    "$$   \\text{rprox}_{\\gamma F}(x) = 2\\text{prox}_{\\gamma F}(x)-x $$\n",
    "\n",
    "\n",
    "It is of course possible to inter-change the roles of $F$ and $G$,\n",
    "which defines another set of iterations.\n",
    "\n",
    "\n",
    "One can show that for any value of $\\gamma>0$, any $ 0 < \\mu < 2 $,\n",
    "and any $\\tilde x_0$, $x_k \\rightarrow x^\\star$\n",
    "which is a minimizer of the minimization of $F+G$.\n",
    "\n",
    "Compressed Sensing Acquisition\n",
    "------------------------------\n",
    "Compressed sensing acquisition corresponds to a random projection\n",
    "$y=Ax_0$ of a signal $x_0$ on a\n",
    "few linear vectors (the lines of $A$). For the recovery of $x_0$ to be possible, it is assumed\n",
    "to be sparsely represented in an orthogonal basis. Up to a change of\n",
    "basis, we suppose that the vector $x$ itself is sparse.\n",
    "\n",
    "\n",
    "Initialize the random number generator to have always the same signals."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
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   "outputs": [],
   "source": [
    "set.seed(0)"
   ]
  },
  {
   "cell_type": "markdown",
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   },
   "source": [
    "Dimension of the problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
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   "outputs": [],
   "source": [
    "n = 400"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Number of measurements."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "deletable": true,
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   "outputs": [],
   "source": [
    "p = round(n/4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Create a random Gaussian measurement matrix $A$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "deletable": true,
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   "outputs": [],
   "source": [
    "A = randn(p,n) / sqrt(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Sparsity of the signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
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   "outputs": [],
   "source": [
    "s = 17"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    "editable": true
   },
   "source": [
    "We begin by generating a $s$-sparse signal $x_0$ with $s$ randomized values.\n",
    "Since the measurement matrix is random, one does not care about the sign\n",
    "of the Diracs, so we use +1 values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
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    "deletable": true,
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   },
   "outputs": [],
   "source": [
    "sel = sample(c(1 : n))\n",
    "x0 = zeros(n, 1)\n",
    "x0[sel[1:s]] = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We perform random measurements $y=Ax_0$ without noise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
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   "outputs": [],
   "source": [
    "y = A %*% x0"
   ]
  },
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   },
   "source": [
    "Compressed Sensing Recovery using DR\n",
    "------------------------------------\n",
    "Compressed sensing recovery corresponds\n",
    "to solving the inverse problem $y=A x_0$, which is ill posed because\n",
    "$x_0$ is\n",
    "higher dimensional than $y$.\n",
    "\n",
    "\n",
    "The reconstruction can be perform with $\\ell^1$ minimization,\n",
    "which regularizes the problems by using the sparsity of the solution.\n",
    "$$ x^\\star \\in \\uargmin{ A x = y } \\norm{x}_1 $$\n",
    "where the $\\ell^1$ norm is defined as\n",
    "$$ \\norm{x}_1 = \\sum_i \\abs{x_i}. $$\n",
    "\n",
    "\n",
    "This is the minimization of a non-smooth function under affine\n",
    "constraints. This can be shown to be equivalent to a linear programming\n",
    "problem, for wich various algorithms can be used (simplex, interior\n",
    "points). We propose here to use the DR iterative algorithm.\n",
    "\n",
    "\n",
    "It is possible to recast this problem as the minimization of $F+G$\n",
    "where $G(x) = \\norm{x}_1$ and $F(x)=\\iota_{H}$ where $H =\n",
    "\\enscond{x}{Ax=y}$ is an affine space, and $\\iota_H$ is the indicator\n",
    "function\n",
    "$$ \\iota_H(x) = \\choice{ 0 \\qifq x \\in H, \\\\ +\\infty \\qifq x \\notin H. } $$\n",
    "\n",
    "\n",
    "The proximal operator of the $\\ell^1$ norm is the soft thresholding:\n",
    "$$ \\text{prox}_{\\gamma \\norm{\\cdot}_1}(x)_i = \\max\\pa{ 0, 1-\\frac{\\ga}{\\abs{x_i}} } x_i. $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "proxG = function(x, gamma) {x * pmax(1 - ((gamma) / abs(x)), 0)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display the 1-D curve of the thresholding."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "image/png": 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Tk5Bw8eXLFixeOPP37s2LGwsLCtW7daHba0vvxS7dppzBg1bKikJM2dSwsCgFvZ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\n/x9Yd1/OAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=5, repr.plot.height=3.5)\n",
    "t = linspace(-1,1)\n",
    "plot(t, proxG(t, 0.3), type=\"l\", col=\"blue\", ylab=\"\", xlab=\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The proximal operator of the indicator function of $H$ is the\n",
    "projector, and does not depends on $\\gamma$.\n",
    "$$ \\text{prox}_{\\gamma \\iota_H}(x)_i = \\text{prox}_F(x) = x + A^* (A A^*)^{-1} (y-Ax). $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "pA = t(A) %*% solve(A%*% t(A))\n",
    "proxF = function(x,y){x + pA %*% (y - A %*% x)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "There are two kinds of Douglas-Rachford iterations, depending on wether\n",
    "you first apply the projection or the thresholding.\n",
    "\n",
    "\n",
    "The first algorithm, (DR1), reads:\n",
    "$$ \\tilde x_{k+1} = \\pa{1-\\frac{\\mu}{2}} \\tilde x_k + \\frac{\\mu}{2} \\text{rprox}_F( \\text{rprox}_\\gamma(\\tilde x_k)  )\n",
    "\\qandq x_k = \\text{prox}_\\gamma(\\tilde x_k) $$\n",
    "\n",
    "\n",
    "The second algorithm, (DR2), reads:\n",
    "$$ \\tilde x_{k+1} = \\pa{1-\\frac{\\mu}{2}} \\tilde x_k + \\frac{\\mu}{2} \\text{rprox}_{\\gamma G}( \\text{rprox}_F(\\tilde x_k)  )\n",
    "\\qandq x_k = \\text{Prox}_F(\\tilde x_k) $$\n",
    "\n",
    "\n",
    "The advantage of (DR2) is the the iterates $x_k$ always satisfy\n",
    "$Ax_k=y$, so that one can only keep track of the evolution of the\n",
    "$\\ell^1$ norm during the iterations. We will use only (DR2) in the\n",
    "following.\n",
    "\n",
    "\n",
    "Set the value of $\\mu$ and $\\gamma$.\n",
    "You might consider using your own value to speed up the convergence."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "mu = 1\n",
    "gamma = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Define the rprox operators."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "rproxG = function(x,tau){2 * proxG(x,tau) - x}\n",
    "rproxF = function(x, tau){2 * proxF(x,tau) - x}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Number of iterations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "niter = 500"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 1__\n",
    "\n",
    "Implement the DR iterative algorithm on |niter| iterations.\n",
    "Keep track of the evolution of the $\\ell^1$ norm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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dK4cwcPHuDRI3TvjmLF5Ffq6WHcOLx9m3E2xdOn2L0bbdvKOTejYkXMmJFxBE1u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\nmzaV+0JKW81Z36VLFwC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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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OQ8/fTTzc3NM2bM2Lp167Fjxy5evNjS0mIwGH744Yc9e/b86Ec/qqioSE9P/+KL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\nAAClEYQAAKURhAAApRGEAACl/T/EYfaUaAD4rgAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"Original Signal\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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BpDQkLuueeeUaNGLV++PC8vz+dr1E7PefXVV70ZvGPHjl69emkfZwJAAxYQENCr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\nu7jcAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"Recovered Signal\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "source(\"nt_solutions/optim_4b_dr/exo1.R\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# Insert your code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We display the convergence speed of the $\\ell^1$ norm on the first half iterations, in log\n",
    "scales."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display the original and the recovered signals.\n",
    "Since the original signal is highly sparse, it is perfectly recovered."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 2__\n",
    "\n",
    "Test the recovery of a less sparse signal.\n",
    "What do you observe ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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44dmzlzZkpKSvv27cPCwgIDA6Oiorp165adnb1kyRKdTufD2mbPnl1QUPCrX/3K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",
      "text/plain": [
       "Plot with title \"Original Signal\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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QAASDRIhAAAACQaJEIAAAASDRIhAAAA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\n0oQQgDQhBCDtL7YGJEDUZ/6JAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"Recovered Signal\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "source(\"nt_solutions/optim_4b_dr/exo2.R\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Evaluation of the CS Recovery Probability\n",
    "-----------------------------------------\n",
    "In order to bench in a randomized manner the efficiency of compressed\n",
    "sensing, we compute the probability $p_s$ for a $s$-sparse signal\n",
    "with random non-zero coefficient locations to be recovered by $\\ell^1$\n",
    "minimization.\n",
    "\n",
    "\n",
    "Put formally, if we call $ x^\\star(y) $ the resolution of the $\\ell^1$\n",
    "problem using measurements $y$, then we want to compute with\n",
    "Monte-Carlo sampling\n",
    "$$ p_s = \\mathbb{P}( x^\\star(Ax)=x \\:\\backslash\\: \\norm{x}_0=s ) $$\n",
    "\n",
    "\n",
    "An important feature of the DR algorithm is that it can be run on many\n",
    "signal at once.\n",
    "\n",
    "\n",
    "Number of signals."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "q = 1000"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "List of benched sparsity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "slist = seq(14, 42, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "List of sparsity of each signal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "Slist = slist[mod(c(1 : q), length(slist)) + 1]\n",
    "\n",
    "# Genetate signals so that |x0(:,j)| has sparsity |Slist(j)|.\n",
    "U = rand(n,q)\n",
    "v = c(apply(U,2,sort))\n",
    "v = v[c(0: (q - 1)) * n + Slist]\n",
    "x0 = 1 * (U <= t(matrix(rep(v, n), ncol=n)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Measurements."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "y = A %*% x0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 3__\n",
    "\n",
    "Perform DR on the set of signals |x0|. Note that the proximal mappings\n",
    "operate in parallel on all the signals in |x0|.\n",
    "Each |i|, count the average number |proba(i)|\n",
    "of recovered vectors of sparsity |slist(i)| (up to a given, small, precision)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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YWFwksQut3EL9/oGPVahPe969atq/cR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",
      "text/plain": [
       "Plot with title \"Evaluation of the CS Recovery Probability\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "source(\"nt_solutions/optim_4b_dr/exo3.R\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<script>\n",
    "  $(document).ready(function(){\n",
    "      $('div.prompt').hide();\n",
    "  });\n",
    "</script>"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "3.4.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}