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   "source": [
    "Performance of Sparse Recovery Using L1 Minimization\n",
    "====================================================\n",
    "\n",
    "*Important:* Please read the [installation page](http://gpeyre.github.io/numerical-tours/installation_python/) for details about how to install the toolboxes.\n",
    "$\\newcommand{\\dotp}[2]{\\langle #1, #2 \\rangle}$\n",
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    "$\\newcommand{\\umax}[1]{\\underset{#1}{\\max}\\;}$\n",
    "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n",
    "$\\newcommand{\\uargmin}[1]{\\underset{#1}{argmin}\\;}$\n",
    "$\\newcommand{\\norm}[1]{\\|#1\\|}$\n",
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    "$\\newcommand{\\choice}[1]{ \\left\\{  \\begin{array}{l} #1 \\end{array} \\right. }$\n",
    "$\\newcommand{\\pa}[1]{\\left(#1\\right)}$\n",
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  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "This tour explores theoritical garantees for the performance of recovery\n",
    "using $\\ell^1$ minimization."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "options(warn=-1) # turns off warnings, to turn on: \"options(warn=0)\"\n",
    "library(pracma)\n",
    "library(imager)\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",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Sparse $\\ell^1$ Recovery\n",
    "--------------------------\n",
    "We consider the inverse problem of estimating an unknown signal $x_0 \\in\n",
    "\\RR^N$ from noisy measurements $y=\\Phi x_0 + w \\in \\RR^P$ where $\\Phi \\in \\RR^{P \\times N}$\n",
    "is a measurement matrix with $P \\leq N$, and $w$ is some noise.\n",
    "\n",
    "\n",
    "This tour is focused on recovery using $\\ell^1$ minimization\n",
    "$$ x^\\star \\in \\uargmin{x \\in \\RR^N} \\frac{1}{2}\\norm{y-\\Phi x}^2 + \\la \\norm{x}_1. $$\n",
    "\n",
    "\n",
    "Where there is no noise, we consider the problem $ \\Pp(y) $\n",
    "$$ x^\\star \\in \\uargmin{\\Phi x = y} \\norm{x}_1. $$\n",
    "\n",
    "\n",
    "We are not concerned here about the actual way to solve this convex\n",
    "problem (see the other numerical tours on sparse regularization) but\n",
    "rather on the theoritical analysis of wether $x^\\star$ is close to\n",
    "$x_0$.\n",
    "\n",
    "\n",
    "More precisely, we consider the following three key properties\n",
    "\n",
    "\n",
    "* *Noiseless identifiability*: $x_0$ is the unique solution of $\n",
    "\\Pp(y) $ for $y=\\Phi x_0$.\n",
    "* *Robustess to small noise*: one has $\\norm{x^\\star - x_0} =\n",
    "O(\\norm{w})$ for $y=\\Phi x_0+w$ if $\\norm{w}$ is smaller than\n",
    "an arbitrary small constant that depends on $x_0$ if $\\la$ is well chosen according to $\\norm{w}$.\n",
    "* *Robustess to bounded noise:* same as above, but $\\norm{w}$ can be\n",
    "arbitrary.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "Note that noise robustness implies identifiability, but the converse\n",
    "is not true in general.\n",
    "\n",
    "\n",
    "Coherence Criteria\n",
    "------------------\n",
    "The simplest criteria for identifiality are based on the coherence of the\n",
    "matrix $\\Phi$ and depends only on the sparsity $\\norm{x_0}_0$ of the\n",
    "original signal. This criteria is thus not very precise and gives very pessimistic\n",
    "bounds.\n",
    "\n",
    "\n",
    "The coherence of the matrix $\\Phi = ( \\phi_i )_{i=1}^N \\in \\RR^{P \\times\n",
    "N}$ with unit norm colum $\\norm{\\phi_i}=1$ is\n",
    "$$ \\mu(\\Phi) = \\umax{i \\neq j} \\abs{\\dotp{\\phi_i}{\\phi_j}}. $$\n",
    "\n",
    "\n",
    "\n",
    "Compute the correlation matrix (remove the diagonal of 1's)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "remove_diag = function(C){C - base::diag(base::diag(C))}\n",
    "Correlation = function(Phi){remove_diag(abs(t(Phi) %*% Phi))}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Compute the coherence $\\mu(\\Phi)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "mu = function(Phi){max(Correlation(Phi))}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The condition\n",
    "$$ \\norm{x_0}_0 < \\frac{1}{2}\\pa{1 + \\frac{1}{\\mu(\\Phi)}} $$\n",
    "implies that $x_0$ is identifiable, and also implies to robustess to small and bounded noise.\n",
    "\n",
    "\n",
    "Equivalently, this condition can be written as $\\text{Coh}(\\norm{x_0}_0)<1$\n",
    "where\n",
    "$$ \\text{Coh}(k) = \\frac{k \\mu(\\Phi)}{ 1 - (k-1)\\mu(\\Phi) } $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "Coh = function(Phi, k){(k * mu(Phi)) / (1 - (k-1) * mu(Phi))}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Generate a matrix with random unit columns in $\\RR^P$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "normalize = function(Phi){t(t(Phi) / sqrt(apply(Phi**2,2, sum)))}\n",
    "PhiRand = function(P, N){normalize(matrix( rnorm(P*N,mean=0,sd=1), P, N))}\n",
    "Phi = PhiRand(250, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Compute the coherence and the maximum possible sparsity to ensure\n",
    "recovery using the coherence bound."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] \"Coherence: 0.32\"\n",
      "[1] \"Sparsity max: 2\"\n"
     ]
    }
   ],
   "source": [
    "c = mu(Phi)\n",
    "print(paste(\"Coherence:\", round(c, 2)))\n",
    "print(paste(\"Sparsity max:\", floor(1/2*(1 + 1/c))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 1__\n",
    "\n",
    "Display how the average coherence of a random matrix\n",
    "decays with the redundancy $\\eta = P/N$ of\n",
    "the matrix $\\Phi$. Can you derive an empirical law between\n",
    "$P$ and the maximal sparsity?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
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qbRmQCgxHCTIkTBbDYNHKgfftADD2jS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\nRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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FhycnLLli2NloqIqm73bkybhpwc+Phg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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "source(\"nt_solutions/sparsity_6_l1_recovery/exo1.R\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "##Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Support and Sign-based Criteria\n",
    "-------------------------------\n",
    "In the following we will consider the support\n",
    "$$ \\text{supp}(x_0) = \\enscond{i}{x_0(i) \\neq 0} $$\n",
    "of the vector $x_0$. The co-support is its complementary $I^c$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "where = function(I)\n",
    "       {\n",
    "         n = length(I)\n",
    "         out = c()\n",
    "         for (i in 1:n)\n",
    "         {\n",
    "             if (I[i] == TRUE)\n",
    "             {\n",
    "                 out = c(out, i)\n",
    "             }\n",
    "         }\n",
    "        return(out)\n",
    "        }\n",
    "\n",
    "supp   = function(s){where(abs(s) > 1e-5)}\n",
    "cosupp = function(s){where(abs(s) < 1e-5)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Given some support $ I \\subset \\{0,\\ldots,N-1\\} $, we will denote as\n",
    "$ \\Phi = (\\phi_m)_{m \\in I} \\in \\RR^{N \\times \\abs{I}}$ the\n",
    "sub-matrix extracted from $\\Phi$ using the columns indexed by $I$.\n",
    "\n",
    "\n",
    "J.J. Fuchs introduces a criteria $F$ for identifiability that depends on the\n",
    "sign of $x_0$.\n",
    "\n",
    "\n",
    "J.J. Fuchs. _Recovery of exact sparse representations in the presence of\n",
    "bounded noise._ IEEE Trans. Inform. Theory, 51(10), p. 3601-3608, 2005\n",
    "\n",
    "\n",
    "Under the condition that $\\Phi_I$ has full rank, the $F$ measure\n",
    "of a sign vector $s \\in \\{+1,0,-1\\}^N$ with $\\text{supp}(s)=I$ reads\n",
    "$$ \\text{F}(s) = \\norm{ \\Psi_I s_I }_\\infty\n",
    "      \\qwhereq \\Psi_I = \\Phi_{I^c}^* \\Phi_I^{+,*} $$\n",
    "where $ A^+ = (A^* A)^{-1} A^* $ is the pseudo inverse of a\n",
    "matrix $A$.\n",
    "\n",
    "\n",
    "The condition\n",
    "$$ \\text{F}(\\text{sign}(x_0))<1 $$\n",
    "implies that $x_0$ is identifiable, and also implies to robustess to\n",
    "small noise. It does not however imply robustess to a bounded noise.\n",
    "\n",
    "\n",
    "Compute $\\Psi_I$ matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "PsiI = function(Phi,I)\n",
    "{\n",
    "keep_indices = c()\n",
    "for (i in 1:dim(Phi)[2])\n",
    "{\n",
    "    if (!(i %in% I))\n",
    "    {\n",
    "        keep_indices = c(keep_indices, i)\n",
    "    }\n",
    "}\n",
    "return (t(Phi[,keep_indices]) %*% t(pinv(as.matrix(Phi[,I]))))\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Compute $\\text{F}(s)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "F = function(Phi,s){base::norm(PsiI(Phi, supp(s)) %*% s[supp(s)], type=\"I\")}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The Exact Recovery Criterion (ERC) of a support $I$,\n",
    "introduced by Tropp in\n",
    "\n",
    "\n",
    "J. A. Tropp. _Just relax: Convex programming methods for identifying\n",
    "sparse signals._ IEEE Trans. Inform. Theory, vol. 52, num. 3, pp. 1030-1051, Mar. 2006.\n",
    "\n",
    "\n",
    "Under the condition that $\\Phi_I$ has full rank, this condition reads\n",
    "$$ \\text{ERC}(I) = \\norm{\\Psi_{I}}_{\\infty,\\infty}\n",
    "      =  \\umax{j \\in I^c} \\norm{ \\Phi_I^+ \\phi_j }_1. $$\n",
    "where $\\norm{A}_{\\infty,\\infty}$ is the $\\ell^\\infty-\\ell^\\infty$\n",
    "operator norm of a matrix $A$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "erc = function(Phi, I){base::norm(PsiI(Phi, I), type=\"I\")}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The condition\n",
    "$$ \\text{ERC}(\\text{supp}(x_0))<1 $$\n",
    "implies that $x_0$ is identifiable, and also implies to robustess to\n",
    "small and bounded noise.\n",
    "\n",
    "\n",
    "One can prove that the ERC is the maximum of the F criterion for all signs of the given\n",
    "support\n",
    "$$ \\text{ERC}(I) = \\umax{ s, \\text{supp}(s) \\subset I } \\text{F}(s). $$\n",
    "\n",
    "\n",
    "The weak-ERC is an approximation of the ERC using only the correlation\n",
    "matrix\n",
    "$$ \\text{w-ERC}(I) = \\frac{\n",
    "      \\umax{j \\in I^c} \\sum_{i \\in I} \\abs{\\dotp{\\phi_i}{\\phi_j}}\n",
    " }{\n",
    "      1-\\umax{j \\in I} \\sum_{i \\neq j \\in I} \\abs{\\dotp{\\phi_i}{\\phi_j}}\n",
    " }$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "setdiff = function(n, I)\n",
    "          {keep_indices = c()\n",
    "            for (i in 1:n)\n",
    "            {\n",
    "                if (!(i %in% I))\n",
    "                {\n",
    "                    keep_indices = c(keep_indices, i)\n",
    "                }\n",
    "            }\n",
    "           return(keep_indices)\n",
    "          }\n",
    "\n",
    "g = function(C,I){apply(as.matrix(C[,I]), 1, sum)}\n",
    "werc_g = function(g,I,J){max(g[J])/(1 - max(g[I]))}\n",
    "werc = function(Phi,I){werc_g(g(Correlation(Phi), I), I, setdiff(dim(Phi)[2], I))}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "One has, if $\\text{w-ERC}(I)>0$,  for $ I = \\text{supp}(s) $,\n",
    "$$ \\text{F}(s) \\leq \\text{ERC}(I) \\leq \\text{w-ERC}(I) \\leq\n",
    "      \\text{Coh}(\\abs{I}). $$\n",
    "\n",
    "\n",
    "This shows in particular that the condition\n",
    "$$ \\text{w-ERC}(\\text{supp}(x_0))<1 $$\n",
    "implies identifiability and robustess to small and bounded noise."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 2__\n",
    "\n",
    "Show that this inequality holds on a given matrix.\n",
    "What can you conclude about the sharpness of these criteria ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] \"N = 2000 , P = 1990 , |I| = 6\"\n",
      "[1] \"F(s) = 0.19\"\n",
      "[1] \"ERC(I) = 0.23\"\n",
      "[1] \"w-ERC(s) = 0.27\"\n",
      "[1] \"Coh(|s|) = 1.65\"\n"
     ]
    }
   ],
   "source": [
    "source(\"nt_solutions/sparsity_6_l1_recovery/exo2.R\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "## Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 3__\n",
    "\n",
    "For a given matrix $\\Phi$ generated using PhiRand, draw as a function of the sparsity $k$\n",
    "the probability that a random sign vector $s$ of sparsity\n",
    "$\\norm{s}_0=k$ satisfies the conditions $\\text{F}(x_0)<1$,\n",
    "$\\text{ERC}(x_0)<1$ and $\\text{w-ERC}(x_0)<1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
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atsVdnDxuNIOs+fKOvUanTsiHXrcOMG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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "source(\"nt_solutions/sparsity_6_l1_recovery/exo3.R\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "#Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Restricted Isometry Criteria\n",
    "----------------------------\n",
    "The restricted isometry constants $\\de_k^1,\\de_k^2$ of a matrix $\\Phi$ are the\n",
    "smallest $\\de^1,\\de^2$ that satisfy\n",
    "$$ \\forall x \\in \\RR^N, \\quad \\norm{x}_0 \\leq k \\qarrq\n",
    "      (1-\\de^1)\\norm{x}^2 \\leq \\norm{\\Phi x}^2 \\leq (1+\\de^2)\\norm{x}^2.  $$\n",
    "\n",
    "\n",
    "E. Candes shows in\n",
    "\n",
    "\n",
    "E. J. Cand s. _The restricted isometry property and its implications for\n",
    "compressed sensing_. Compte Rendus de l'Academie des Sciences, Paris, Serie I, 346 589-592\n",
    "\n",
    "\n",
    "that if\n",
    "$$ \\de_{2k} \\leq \\sqrt{2}-1 ,$$\n",
    "then $\\norm{x_0} \\leq k$ implies identifiability as well as robustness to small and bounded noise.\n",
    "\n",
    "\n",
    "The stability constant $\\la^1(A), \\la^2(A)$ of a matrix\n",
    "$A = \\Phi_I$ extracted from $\\Phi$ is the smallest $\\tilde \\la_1,\\tilde \\la_2$ such that\n",
    "$$ \\forall \\al \\in \\RR^{\\abs{I}}, \\quad\n",
    "      (1-\\tilde\\la_1)\\norm{\\al}^2 \\leq \\norm{A \\al}^2 \\leq (1+\\tilde \\la_2)\\norm{\\al}^2.  $$\n",
    "\n",
    "\n",
    "These constants $\\la^1(A), \\la^2(A)$ are easily computed from the\n",
    "largest and smallest eigenvalues of $A^* A \\in \\RR^{\\abs{I} \\times \\abs{I}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "minmax = function(v){c(1 - min(v), max(v) - 1)}\n",
    "ric = function(A){minmax(eigen(t(A) %*% A)$values)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The restricted isometry constant of $\\Phi$ are computed as the largest\n",
    "stability constants of extracted matrices\n",
    "$$ \\de^\\ell_k = \\umax{ \\abs{I}=k } \\la^\\ell( \\Phi_I ).  $$\n",
    "\n",
    "\n",
    "The eigenvalues of $\\Phi$ are essentially contained in the\n",
    "interval $ [a,b] $ where $a=(1-\\sqrt{\\be})^2$ and $b=(1+\\sqrt{\\be})^2$\n",
    "with $\\beta = k/P$\n",
    "More precisely, as $k=\\be P$ tends to infinity, the distribution of the\n",
    "eigenvalues tends to the Marcenko-Pastur law\n",
    "$ f_\\be(\\la) = \\frac{1}{2\\pi \\be \\la}\\sqrt{ (\\la-b)^+ (a-\\la)^+ }. $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 4__\n",
    "\n",
    "Display, for an increasing value of $k$ the histogram of repartition\n",
    "of the eigenvalues $A^* A$ where $A$ is a Gaussian matrix of size $(P,k)$ and\n",
    "variance $1/P$. For this, accumulate the eigenvalues for many\n",
    "realizations of $A$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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WV2DsAnUDjgA86cEYUDhbVpo7AwLV2qgwfNjgJ4PwoHfMDZs5LOmJ0CnmjsWOXk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\nAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"P= 200 , k= 10\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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7ZNv01kZKQkd3fmu2TXTz9dvv/+C3/9ZXYdcHH2hc9vaMBCNVgo1dKxPtrQWd/M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\nRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"P= 200 , k= 30\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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VznQlSb6UjgkEgkAORyudiFaDdLPGmCBxFwF7sQ0lNl1rk8A/YB+wAAXRHhhs0f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\nusdOLQuHyQsAAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"P= 200 , k= 50\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "source(\"nt_solutions/sparsity_6_l1_recovery/exo4.R\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "## Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 5__\n",
    "\n",
    "Estimate numerically lower bound on $\\de_k^1,\\de_k^2$ by Monte-Carlo\n",
    "sampling of sub-matrices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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LyiHaMTnRL9TPzultz1Tfa1+8NuWfqy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\nAACdoNgBAADoxP8HOHiYGR+HRW8AAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"N= 600 , P= 300\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "source(\"nt_solutions/sparsity_6_l1_recovery/exo5.R\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "## Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Sparse Spikes Deconvolution\n",
    "---------------------------\n",
    "We now consider a convolution dictionary $ \\Phi $.\n",
    "Such a dictionary is used with sparse regulariz\n",
    "\n",
    "\n",
    "Second derivative of Gaussian kernel $g$ with a given variance $\\si^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "sigma = 6\n",
    "g = function(x){(1 - (x**2 / sigma**2)) * exp(-x**2/(2*sigma**2))}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Create a matrix $\\Phi$ so that $\\Phi x = g \\star x$ with periodic\n",
    "boundary conditions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "P = 1024\n",
    "t = meshgrid(0:(P - 1),0:(P - 1))\n",
    "Y = t$X\n",
    "X = t$Y\n",
    "Phi = normalize(g((X - Y + P/2.) %% P - P/2.))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "To improve the conditionning of the dictionary, we sub-sample its atoms,\n",
    "so that $ P = \\eta N > N $."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "eta = 2.\n",
    "N = P/eta\n",
    "Phi = Phi[,seq(1, dim(Phi)[2], by=eta)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Plot the correlation function associated to the filter.\n",
    "Can you determine the value of the coherence $\\mu(\\Phi)$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ATsh4oQsBBBCNgPzwgBCxGEgM00NMjv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\nAHA1ghAA4Gr/H1SFkW8R6jSDAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "c = t(Phi) %*% Phi\n",
    "c = abs(c[,dim(c)[2]/2])\n",
    "\n",
    "options(repr.plot.width=5, repr.plot.height=5)\n",
    "plot(c[(length(c)/2 - 50):((length(c)/2) + 50)], type=\"l\", xlab=\"\", ylab=\"\", col=4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "roll <- function( x , n ){\n",
    "  if( n == 0 )\n",
    "    return( x )\n",
    "  c( tail(x,n) , head(x,-n) )\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "twosparse = function(d){roll(c(1, rep(0, d), -1, rep(0, N - d- 2)), as.integer(N/2 - d/2))}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display $x_0$ and $\\Phi x_0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "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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PKA1fLzRUS6ugZfxt3nOm5qtKtLrr764sRpyiVOjXrsN08EjAlBCFhNBWF39+DL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\n",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x0 = twosparse(50)\n",
    "\n",
    "plot(x0, type=\"l\", col=2, ylab=\"\", xlab=\"\")\n",
    "plot(Phi %*% x0, type=\"l\", col=4, ylab=\"\", xlab=\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 6__\n",
    "\n",
    "Plot the evolution of the criteria F, ERC and Coh as a function of $d$.\n",
    "Do the same plot for other signs patterns for $x_0$.\n",
    "Do the same plot for a Dirac comb with a varying spacing $d$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Bg4G0tERDUEi5042j58IilJ8JJYmb8/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\ngg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "source(\"nt_solutions/sparsity_6_l1_recovery/exo6.R\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "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": {
  "anaconda-cloud": {},
  "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": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}