{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "Author: V. Cottet\n", "Date: 02/06/2015\n", "\n", "This notebook aims at doing Matrix completion under Quantile Loss with Nuclear norm penalization. The program is:\n", "$$ \n", "\\widehat{M} = \\textrm{arg}\\min_M \\frac{1}{N} \\sum_{i=1}^{N} \\left(\\tau (Y_i-M_{X_i})_+ + (1-\\tau)(M_{X_i}-Y_i)_+ \\right) + \\mu ||M||_{S_1} \\tag{QN}\n", "$$\n", "\n", "The first part is to create the examples, the second is the main function by ADMM with parameters (Y_obs,indices, tau, mu,rho, tol), rho and tol are parameters required by the optimization method.\n", "\n", "# Section 1: Matrix creation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Initialization " ] }, { "cell_type": "code", "execution_count": 85, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Init the prog\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import cvxpy as cvx\n", "from scipy.sparse.linalg import svds # Fast SVD for sparse matrix using ARPACK Fortran routines\n", "from __future__ import division\n", "%matplotlib inline\n", "\n", "np.random.seed(1000)\n", "\n", "# Init the general parameters\n", "rank = 3\n", "m1 = 200\n", "m2 = 200\n", "rate = .2\n", "n = int(m1*m2*rate)\n", "\n", "# Define the test functions\n", "def SquaredLossTot(Y,M):\n", " result = np.mean((Y-M)**2)\n", " return(result)\n", "\n", "def AbsLossTot(Y,M):\n", " result = np.mean(abs(Y-M))\n", " return(result)\n", "\n", "def SquaredLossTest(Yobs,M,indices):\n", " result = np.mean((Yobs-M.flat[indices])**2)\n", " return(result)\n", "\n", "def AbsLossTest(Yobs,M,indices):\n", " result = np.mean(abs(Yobs-M.flat[indices]))\n", " return(result)" ] }, { "cell_type": "code", "execution_count": 86, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Matrix Creation Functions\n", "# Type A: Gaussian noise \n", "def AbsMatrixCreationA(m1,m2,rank,n,lnoise):\n", " L = np.random.normal(0,1,size=(m1,rank))\n", " R = np.random.normal(0,1,size=(rank,m2))\n", " Y = np.dot(L,R)\n", " indices = np.random.choice(m1*m2,n)\n", " Yobs = Y.flat[indices] + np.sqrt(lnoise)*np.random.normal(0,1,size=n) \n", " return(Y,Yobs,indices)\n", "\n", "# Type B: Student noise, df to be chosen\n", "def AbsMatrixCreationB(m1,m2,rank,n,lnoise,df):\n", " L = np.random.normal(0,1,size=(m1,rank))\n", " R = np.random.normal(0,1,size=(rank,m2))\n", " Y = np.dot(L,R)\n", " indices = np.random.choice(m1*m2,n)\n", " Yobs = Y.flat[indices] + np.sqrt(lnoise)*np.random.standard_t(df=df,size=n) \n", " return(Y,Yobs,indices)\n", "\n", "# Type C: Gaussian noise with outliers (about 10 shift)\n", "def AbsMatrixCreationC(m1,m2,rank,n,lnoise,pct_outlier):\n", " L = np.random.normal(0,1,size=(m1,rank))\n", " R = np.random.normal(0,1,size=(rank,m2))\n", " Y = np.dot(L,R)\n", " indices = np.random.choice(m1*m2,n)\n", " outlier = np.random.binomial(1,pct_outlier,n)\n", " sign = 2*np.random.binomial(1,.5,n)-1\n", " Yobs = Y.flat[indices] + np.sqrt(lnoise)*np.random.normal(0,1,n)+outlier*sign*10 \n", " return(Y,Yobs,indices)\n" ] }, { "cell_type": "code", "execution_count": 87, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "(array([ 1.00000000e+00, 1.00000000e+00, 1.00000000e+00,\n", " 4.00000000e+00, 9.00000000e+00, 2.70000000e+01,\n", " 9.60000000e+01, 3.17000000e+02, 7.55000000e+02,\n", " 2.09400000e+03, 5.15600000e+03, 1.09200000e+04,\n", " 1.14730000e+04, 5.61900000e+03, 2.22700000e+03,\n", " 8.40000000e+02, 3.12000000e+02, 1.05000000e+02,\n", " 3.00000000e+01, 1.30000000e+01]),\n", " array([-13.35607568, -12.24778839, -11.1395011 , -10.03121381,\n", " -8.92292652, -7.81463924, -6.70635195, -5.59806466,\n", " -4.48977737, -3.38149008, -2.27320279, -1.1649155 ,\n", " -0.05662821, 1.05165907, 2.15994636, 3.26823365,\n", " 4.37652094, 5.48480823, 6.59309552, 7.70138281, 8.80967009]),\n", " )" ] }, "execution_count": 87, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Test\n", "[Y1,Yobs1,indices1]=AbsMatrixCreationA(m1,m2,rank,n,0)\n", "[Y2,Yobs2,indices2]=AbsMatrixCreationB(m1,m2,rank,n,1,3)\n", "[Y3,Yobs3,indices3]=AbsMatrixCreationC(m1,m2,rank,n,1,.2)\n", "plt.hist(np.reshape(Y1,m1*m2,1),20)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Section 2: Quantile Completion Functions\n", "## Section 2.1: SDP Algorithm\n", "\n", "By introducing Slack Variables, the (QN) problem is equivalent to:\n", "\\begin{eqnarray}\n", "\\left(\\hat{X},\\hat{Z},\\hat{U},\\hat{V} \\right) =\n", "& \\textrm{argmin} &\\frac{1}{N}\\sum (X_i+Z_i) + \\mu\\frac{\\textrm{Tr}(U) + \\textrm{Tr}(V)}{2}\\\\\n", " & \\textrm{subject to:} &\\left[ \\begin{array} UU & M \\\\\n", " M^\\top & V \\end{array} \\right] \\succeq 0 \\\\\n", " && X_i \\geq 0 \\wedge \\tau(Y_i-M_{X_i}) \\\\\n", " && Z_i \\geq 0 \\wedge (1-\\tau)(M_{X_i}-Y_i) \\tag{SDP}\n", "\\end{eqnarray}\n", "\n", "Actually, the solver does it by itself so we just had the slack variables and therefore the problem (QN) is equivalent to:\n", "\\begin{eqnarray}\n", "\\left(\\hat{X},\\hat{Z},\\hat{M} \\right) =\n", "& \\textrm{argmin} &\\frac{1}{N}\\sum (X_i+Z_i) + \\mu||M||_{S_1}\\\\\n", " & \\textrm{subject to:} & X_i \\geq 0 \\wedge \\tau(Y_i-M_{X_i}) \\\\\n", " && Z_i \\geq 0 \\wedge (1-\\tau)(M_{X_i}-Y_i) \\tag{Slack}\n", "\\end{eqnarray}" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def QuantileMatCompletionSDP(Yobs,indices,m1,m2,tau,mu):\n", " n = len(Yobs)\n", " X = cvx.Variable(n)\n", " Z = cvx.Variable(n)\n", " M = cvx.Variable(m1,m2)\n", " constr1 = (X >= 0)\n", " constr2 = (Z >= 0)\n", " constr3 = (X - tau*(Yobs-cvx.vec(M.T)[indices]) >= 0)\n", " constr4 = (Z - (1-tau)*(cvx.vec(M.T)[indices]-Yobs) >= 0)\n", " obj = cvx.Minimize(sum(X)+sum(Z)+mu*cvx.norm(M,'nuc'))\n", "\n", " prob = cvx.Problem(obj,[constr1,constr2,constr3,constr4])\n", " prob.solve(solver=\"SCS\")\n", " return(np.array(M.value))\n", "\n", "[Y4,Yobs4,indices4]=AbsMatrixCreationA(20,20,2,200,0)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "A = QuantileMatCompletionSDP(Yobs4,indices4,20,20,.5,3)\n", "plt.plot(np.linalg.svd(A)[1])" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "A_sdp = QuantileMatCompletionSDP(Yobs1,indices1,200,200,.5,4)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "print(SquaredLossTest(Yobs1,A_sdp,indices1), SquaredLossTot(Y1,A_sdp), AbsLossTest(Yobs1,A_sdp,indices1), AbsLossTot(Y1,A_sdp))\n", "print(\"la norme nucléaire est : \")\n", "print(sum(np.linalg.svd(A_sdp)[1]))\n", "plt.figure(figsize=(15,4))\n", "plt.subplot(1,2,1)\n", "plt.hist(np.reshape(abs(Y1-A_sdp),m1*m2,1), 20, lw=2)\n", "plt.title(\"Distribution of the gaps\")\n", "plt.subplot(1,2,2)\n", "plt.plot(np.linalg.svd(A_sdp)[1][0:11], lw=2)\n", "plt.title(\"10 first Singular values of M\")\n", "plt.xlim([-2, 12])" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "A2_sdp = QuantileMatCompletionSDP(Yobs1,indices1,200,200,.5,5)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "print(SquaredLossTest(Yobs1,A2_sdp,indices1), SquaredLossTot(Y1,A2_sdp), AbsLossTest(Yobs1,A2_sdp,indices1), AbsLossTot(Y1,A2_sdp))\n", "print(\"la norme nucléaire est : \")\n", "print(sum(np.linalg.svd(A2_sdp)[1]))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Section 2.2: ADMM Algorithm\n", "\n", "The ADMM Algorithm is used with the following program:\n", "$$ \n", "\\min f(M) + g(N) \\\\\n", "\\textrm{s.c.} M=N\n", "$$\n", "\n", "The updates are then:\n", "$$\n", "M^k = \\textrm{argmin}_M \\quad f(M) + \\frac{\\rho}{2} ||M-N^{k-1}+U^{k-1}||^2_F =\\textrm{prox}_{f/\\rho} (N^{k-1}-U^{k-1}) \\\\\n", "N^k = \\textrm{argmin}_N \\quad g(N) + \\frac{\\rho}{2} ||M^k-N+U^{k-1}||^2_F =\\textrm{prox}_{g/\\rho} (M^k+U^{k-1})\\\\\n", "U^k = U^{k-1} + M^k-N^k \n", "$$\n", "\n", "Here $g(N) = \\mu ||N||_{S_1}$ and $f(M)=\\frac{1}{N} \\sum_{i=1}^{N} \\left(\\tau (Y_i-M_{X_i})_+ + (1-\\tau)(M_{X_i}-Y_i)_+ \\right)$. $\\textrm{prox}_{g/\\rho}$ corresponds to the soft-thresholding of the singular values with parameter $\\mu/\\rho$. \n" ] }, { "cell_type": "code", "execution_count": 88, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def QuantileMatCompletionADMM(Yobs, indices, m1,m2, tau, mu, rho,tol,itermax=1000):\n", " n = len(Yobs)\n", " M = np.random.normal(0,1,size=(m1,m2))\n", " N = np.zeros((m1,m2))\n", " U = np.zeros((m1,m2))\n", " \n", " # All the variables to float type:\n", " mu = float(mu) ; rho = float(rho)\n", " \n", " if n != len(indices): print(\"There is a big Problem, check Yobs and indices\")\n", " \n", " # function to evaluate the minimum of:\n", " # tau(y-x)_+ + (1-tau)(x-y)_+ + r/2(x-b)^2\n", " # w.r.t x\n", " \n", " def min_abs(y,b):\n", " x1 = np.minimum(y,b+tau/rho)\n", " x2 = np.maximum(y,b-(1-tau)/rho)\n", " y1 = tau*(y-x1) + rho/2 * (x1-b)**2\n", " y2 = (1-tau)*(x2-y) + rho/2*(x2-b)**2\n", " ind_min = np.where(y1 x\n", " S[j] = S[j] - x\n", " l=sum(j)\n", " return(np.dot(u[:,:l],np.dot(np.diag(S[:l]),v[:l,:])))\n", " \n", " def update_N(M,U):\n", " return(SoftThresMat(M+U,mu/rho))\n", " \n", " def update_U(M,N,U):\n", " return(U + M -N)\n", " \n", " # Main Loop\n", " \n", " ec = 1000.\n", " k = 0\n", " while(ec>tol and k" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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MVQ4CzqyqR6pqObAMWJxkW2BeVV3eLncqcHDHOqe002cD+03jWCRJkvqOSZ+k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\nRvD0uj5JazJTt2yQJElrKcl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qlcAKYHGSHYEtq+qKdr2zgcM76pzVLp8PvHQa+yJJktR3TPokzbUp3dOX5FnA\nAcC32qLjk1yd5KNJtm7LFgI3d1Rb1ZYtBG7pKL+FJ5LHx+tU1aPAfUmeNpXYJEmS+pHTNkiaawsm\nu2KSLWiuwp1QVQ8kOQP4P1VVSf4C+ADw5lmKa6wriK3TgM3b5aH2IUnqd8PDwwwPD3c7DGneeaVP\n0lybVNKXZAFNwndOVV0AUFV3dqzyT8AX2uVVwM4dr+3Ulo1X3lnn1iQbAltV1T1jR3MisO1kwpYk\n9ZGhoSGGhoYef7506dLuBTOHkpwJvApYXVXPactOAd4C3NGudnLHvfInAW+iucf+hKq6dP6j1lwa\nudL3gx/Ao4/Chht2Nx5Jg2ey3Ts/Biyrqr8ZKWjv0RvxauB77fKFwJHtiJy7AXsCl1fV7cD9SRa3\nA7scBVzQUefodvm1wGXT2htJknrfx4GDxyif6uBoGhBbbw0LF8IvfgErV3Y7GkmDaMIrfUleDPwO\ncG2Sq4ACTgZen+QA4DFgJfBWgKpaluQ8YBnwCHBcVVW7ubcBnwA2BS4aadSAM4FzkqwA7gaOnJW9\nkySpx1TVV5PsOsZL6xwcDVjZtpOLeeLeeg2IRYtg1armvr499uh2NJIGzYRJX1V9DRiro8HFY5SN\n1DkVOHWM8iuB/ccof4jmTKYkSeur45O8Afg28MdVdT/NQGff6FhnZHA0DZj99oMvfam5r+9Vr+p2\nNJIGzZRG75QkSXPiDGD3qjoAuJ1mcDStRxzMRdJcmvTonZIkaW5MY3C0MS1ZsuTx5dED46i3jSR9\nTtsgabTZGN3apE+SpPkXOu7hS7JjO+AZPHlwtE8m+RBNt849gcvH22hn0qf+MjKC57JlUAUO1yNp\nxGyMbm3SJ0nSPEryKZpJZrdNchNwCvDr0xgcTQPk6U+H7baDO++EW26BnXeeuI4kTZZJnyRJ86iq\nXj9G8cfXsf6Yg6Np8Oy3H3zlK83VPpM+SbPJgVwkSZJ6wEgXT+/rkzTbTPokSZJ6gCN4SporJn2S\nJEk9wKRP0lwx6ZMkSeoBnUmfw/VImk0mfZIkST1gxx1h663h3nvhjju6HY2kQWLSJ0mS1AMSu3hK\nmhsmfZIkST3CpE/SXDDpkyRJ6hFO2yBpLpj0SZIk9Qiv9EmaCyZ9kiRJPcKkT9JcMOmTJEnqETvv\nDE95CqxeDffc0+1oJA0Kkz5JkqQescEGsO++zbL39UmaLSZ9kiRJPcQunpJmm0mfJElSDzHpkzTb\nJkz6kuySKs9jAAAX3ElEQVSU5LIk309ybZK3t+XbJLk0yXVJLkmydUedk5KsSLI8yUEd5QcmuSbJ\n9UlO7yjfOMm5bZ1vJNlltndUkiSpHzhtg6TZNpkrfWuAd1TVs4EXAm9Lsi/wLuBLVbUPcBlwEkCS\n/YAjgEXAIcAZSdJu6yPAsVW1N7B3koPb8mOBe6pqL+B04P2zsneSJEl9xit9kmbbhElfVd1eVVe3\nyw8Ay4GdgMOAs9rVzgIOb5cPBc6tqjVVtRJYASxOsiOwZVVd0a53dkedzm2dD7x0JjslSZLUr3bb\nDTbZBG6+GX76025HI2kQTOmeviTPAg4AvgnsUFWroUkMge3b1RYCN3dUW9WWLQRu6Si/pS1bq05V\nPQrcl+RpU4lNkiRpECxYAPvs0yz/4AfdjUXSYFgw2RWTbEFzFe6EqnogSY1aZfTzmcj4L50GbN4u\nD7UPSVK/Gx4eZnh4uNthSD1h0SK45pqmi+fzn9/taCT1u0klfUkW0CR851TVBW3x6iQ7VNXqtuvm\nHW35KmDnjuo7tWXjlXfWuTXJhsBWVTXOlKQnAttOJmxJUh8ZGhpiaGjo8edLly7tXjBSl3lfn6TZ\nNNnunR8DllXV33SUXQgc0y4fDVzQUX5kOyLnbsCewOVtF9D7kyxuB3Y5alSdo9vl19IMDCNJkrRe\nMumTNJsmvNKX5MXA7wDXJrmKphvnyTT9LM9L8ibgRpoRO6mqZUnOA5YBjwDHVdVI18+3AZ8ANgUu\nqqqL2/IzgXOSrADuBo6cnd2TJEnqPyNJn9M2SJoNEyZ9VfU1YMNxXn7ZOHVOBU4do/xKYP8xyh+i\nTRolSZLWd3vuCRtuCD/6Efz857DZZt2OSFI/m9LonZIkSZp7G28Me+0FVXDddd2ORlK/M+mTJEnq\nQXbxlDRbTPokSZJ60KJFzb8O5iJppkz6JEmSepAjeEqaLSZ9kiRJPcikT9JsMemTJEnqQfvsAwn8\n8Ifw8MPdjkZSPzPpkyRJ6kGbbQa77QZr1jSJnyRNl0mfJElSj7KLp6TZYNInSZLUo0z6JM0Gkz5J\nkqQeNTJtg3P1SZoJkz5JkqQe5ZU+SbPBpE+SJKlHjVzpu+46ePTR7sYiqX+Z9EmSJPWoLbeEnXeG\nhx6CH/+429FI6lcmfZIkzaMkZyZZneSajrJtklya5LoklyTZuuO1k5KsSLI8yUHdiVrdNHK1zy6e\nkqbLpE+SpPn1ceDgUWXvAr5UVfsAlwEnASTZDzgCWAQcApyRJPMYq3qA9/VJmimTPkmS5lFVfRW4\nd1TxYcBZ7fJZwOHt8qHAuVW1pqpWAiuAxfMRp3qHSZ+kmTLpkySp+7avqtUAVXU7sH1bvhC4uWO9\nVW2Z1iNO2yBppkz6JEnqPdXtANQ7OpO+xx7rbiyS+tOCbgcgSZJYnWSHqlqdZEfgjrZ8FbBzx3o7\ntWVjWrJkyePLQ0NDDA0NzX6kmnfbbgs77ACrV8PNN8Ouu3Y7IknzaXh4mOHh4RltY8KkL8mZwKuA\n1VX1nLbsFOAtPNEonVxVF7evnQS8CVgDnFBVl7blBwKfADYFLqqqP2zLNwbOBp4H3AW8rqpumtFe\nSZLU29I+RlwIHAOcBhwNXNBR/skkH6Lp1rkncPl4G+1M+jRY9tuvSfqWLzfpk9Y3o0/iLV26dMrb\nmEz3zrFGGQP4YFUd2D5GEr5FjD/K2EeAY6tqb2DvJCPbPBa4p6r2Ak4H3j/lvZAkqU8k+RTwdZq2\n8KYkbwTeB7w8yXXAS9vnVNUy4DxgGXARcFxV2fVzPeS0DZJmYsIrfVX11SRjnVMaa8jow2hHGQNW\nJlkBLE5yI7BlVV3Rrnc2zchkl7R1TmnLzwf+bor7IElS36iq14/z0svGWf9U4NS5i0j9wBE8Jc3E\nTAZyOT7J1Uk+2jGJ7HijjC0Ebukov4UnRh97vE5VPQrcl+RpM4hLkiRpoJj0SZqJ6Q7kcgbwf6qq\nkvwF8AHgzbMU0wSTzp4GbN4uD7UPSVK/m40b1aVB1TmCZxVkgl9LktRpWklfVd3Z8fSfgC+0y+ON\nMrau0cdGXrs1yYbAVlV1z/jvfiKw7XTCliT1sNm4UV0aVDvsANtsA/feC7ffDs94RrcjktRPJtu9\nc61RxtrhpEe8Gvheu3whcGSSjZPsRjvKWDvR7P1JFrcDuxzF2iOTHd0uvxa4bFp7IkmSNKASu3hK\nmr4Jk75xRhl7f5JrklwN/BrwRzDhKGNvA84ErgdWjIz42ZY9vR305Q+Bd83a3kmSJA2IkaRv+fLu\nxiGp/0xm9M6xRhn7+DrWH3OUsaq6Eth/jPKHaKZ5kCRJ0jictkHSdM1k9E5JkiTNE7t3Spoukz5J\nkqQ+YNInabpM+iRJkvrATjvBFlvAnXfCXXd1OxpJ/cSkT5IkqQ8ka8/XJ0mTZdInSZLUJ+ziKWk6\nTPokSZL6hNM2SJoOkz5JkqQ+4bQNkqbDpE+SJKlP2L1T0nSY9EmSJPWJZz0LNt0UVq2C++/vdjSS\n+oVJnyRJUp/YcEPYd99m+Qc/6G4skvqHSZ8kSVIf8b4+SVNl0idJktRHvK9P0lSZ9EmSJPURp22Q\nNFUmfZIkSX3E7p2SpsqkT5IkqY/suScsWAArV8LPftbtaCT1A5M+SZKkPrLRRrD33lAF113X7Wgk\n9QOTPkmSpD7jfX2SpsKkT5Ikqc94X5+kqTDpkyRJ6jNO2yBpKiZM+pKcmWR1kms6yrZJcmmS65Jc\nkmTrjtdOSrIiyfIkB3WUH5jkmiTXJzm9o3zjJOe2db6RZJfZ3EFJkqRBY/dOSVMxmSt9HwcOHlX2\nLuBLVbUPcBlwEkCS/YAjgEXAIcAZSdLW+QhwbFXtDeydZGSbxwL3VNVewOnA+2ewP5IkSQNv771h\ngw3ghz+Ehx7qdjSSet2ESV9VfRW4d1TxYcBZ7fJZwOHt8qHAuVW1pqpWAiuAxUl2BLasqiva9c7u\nqNO5rfOBl05jPyRJktYbm24Ku+8Ojz4KK1Z0OxpJvW669/RtX1WrAarqdmD7tnwhcHPHeqvasoXA\nLR3lt7Rla9WpqkeB+5I8bZpxSZIkrRe8r0/SZC2Ype3ULG0HIOt++TRg83Z5qH1Ikvrd8PAww8PD\n3Q5D6hv77QcXXuh9fZImNt2kb3WSHapqddt18462fBWwc8d6O7Vl45V31rk1yYbAVlV1z/hvfSKw\n7TTDliT1qqGhIYaGhh5/vnTp0u4FI/UBp22QNFmT7d4Z1r4CdyFwTLt8NHBBR/mR7YicuwF7Ape3\nXUDvT7K4HdjlqFF1jm6XX0szMIwkSZLWwe6dkiZrwit9ST5F04dy2yQ3AacA7wM+k+RNwI00I3ZS\nVcuSnAcsAx4Bjquqka6fbwM+AWwKXFRVF7flZwLnJFkB3A0cOTu7JkmSNLj23bf597rrYM0aWDBb\nN+1IGjgTfj1U1evHeell46x/KnDqGOVXAvuPUf4QbdIoSZKkydliC9hlF7jpJvjRj5ppHCRpLNMd\nvVOSJEldZhdPSZNh0idJktSnTPokTYa9vyVJ6hFJVgL3A48Bj1TV4iTbAJ8GdgVWAkdU1f1dC1I9\nZSTpc9oGSevilT5JknrHY8B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KWJJkV2BBVd3QbnchcHTHPsvb5cuAV06jLZIkSX3FB7RLmgtTmtOX5AXAwcB1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\nIsntNIFgp3G/2p9xxhnPLA8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Aio7yFaxJHh/bpqoeAX6eZPupxCZJkjSobOmTNJvmdrtikq1oWuHeVVX3J/kU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\ndqx/XJJlSW5IckhH+YFJrk2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TGq9JCm5net07lwDXs3r1apYsWTLLkUmSZlsSqmo9Y7zVKYnrtveh+++HTdup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\ncE5H+UHtjJy7AUuBS9ouoHc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CbkckaVBNaPZOSZIkTR+HeEqaDRZ9kiRJXTJU9C1d2t04JA02iz5JkqQusadP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\nJEmS1GXbbANPeQqsWgV33tn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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "for mu in np.arange(10):\n", " M,N,U,k = QuantileMatCompletionADMM(Yobs1, indices1, m1,m2, tau=.5, mu=mu, rho=.2, tol=.00001, itermax=2000)\n", " print(mu, k)\n", " print(SquaredLossTest(Yobs1,M,indices1), SquaredLossTot(Y1,M), AbsLossTest(Yobs1,M,indices1), AbsLossTot(Y1,M))\n", " print(\"The Nuclear norm is: \", sum(np.linalg.svd(M)[1]))\n", " plt.figure(figsize=(15,4))\n", " plt.subplot(1,2,1)\n", " plt.hist(abs(np.reshape(Y1-M,m1*m2,1)), 20, lw=2)\n", " plt.title(\"Distribution of the gaps\")\n", " plt.subplot(1,2,2)\n", " plt.plot(np.linalg.svd(M)[1][0:11], lw=2)\n", " plt.title(\"10 first Singular values of M\")\n", " plt.xlim([-2, 12])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Second Test: different values for $\\mu$ with large $\\rho$ (=1). The solutions are always perfect, strange behavior of the algorithm. " ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [], "source": [ "for mu in np.arange(10):\n", " M,N,U,k = QuantileMatCompletionADMM(Yobs1, indices1, m1,m2, tau=.5, mu=mu, rho=1, tol=.00001, itermax=2000)\n", " print(mu, k)\n", " print(SquaredLossTest(Yobs1,M,indices1), SquaredLossTot(Y1,M), AbsLossTest(Yobs1,M,indices1), AbsLossTot(Y1,M))\n", " print(\"The Nuclear norm is: \", sum(np.linalg.svd(M)[1]))\n", " plt.figure(figsize=(15,4))\n", " plt.subplot(1,2,1)\n", " plt.hist(abs(np.reshape(Y1-M,m1*m2,1)), 20, lw=2)\n", " plt.title(\"Distribution of the gaps\")\n", " plt.subplot(1,2,2)\n", " plt.plot(np.linalg.svd(M)[1][0:11], lw=2)\n", " plt.title(\"10 first Singular values of M\")\n", " plt.xlim([-2, 12])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Section 2.2.2: Test with small level of noise" ] }, { "cell_type": "code", "execution_count": 91, "metadata": { "collapsed": true }, "outputs": [], "source": [ "[Y11,Yobs11,indices11]=AbsMatrixCreationA(m1,m2,rank,n,0.2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Small value of $\\rho$ (=0.2)." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "for mu in np.arange(10):\n", " M,N,U,k = QuantileMatCompletionADMM(Yobs11, indices11, m1,m2, tau=.5, mu=mu, rho=.2, tol=.00001, itermax=2000)\n", " print(mu, k)\n", " print(SquaredLossTest(Yobs11,M,indices11), SquaredLossTot(Y11,M), AbsLossTest(Yobs11,M,indices11), AbsLossTot(Y11,M))\n", " print(\"The Nuclear norm is: \", sum(np.linalg.svd(M)[1]))\n", " plt.figure(figsize=(15,4))\n", " plt.subplot(1,2,1)\n", " plt.hist(abs(np.reshape(Y11-M,m1*m2,1)), 20, lw=2)\n", " plt.title(\"Distribution of the gaps\")\n", " plt.subplot(1,2,2)\n", " plt.plot(np.linalg.svd(M)[1][0:11], lw=2)\n", " plt.title(\"10 first Singular values of M\")\n", " plt.xlim([-2, 12])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "large value of $\\rho$ (=1)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "for mu in np.arange(10):\n", " M,N,U,k = QuantileMatCompletionADMM(Yobs11, indices11, m1,m2, tau=.5, mu=mu, rho=1, tol=.00001, itermax=2000)\n", " print(mu, k)\n", " print(SquaredLossTest(Yobs11,M,indices11), SquaredLossTot(Y11,M), AbsLossTest(Yobs11,M,indices11), AbsLossTot(Y11,M))\n", " print(\"The Nuclear norm is: \", sum(np.linalg.svd(M)[1]))\n", " plt.figure(figsize=(15,4))\n", " plt.subplot(1,2,1)\n", " plt.hist(abs(np.reshape(Y11-M,m1*m2,1)), 20, lw=2)\n", " plt.title(\"Distribution of the gaps\")\n", " plt.subplot(1,2,2)\n", " plt.plot(np.linalg.svd(M)[1][0:11], lw=2)\n", " plt.title(\"10 first Singular values of M\")\n", " plt.xlim([-2, 12])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For a fixed $\\mu$, study of the effect of $\\rho$" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "for rho in np.arange(10):\n", " M,N,U,k = QuantileMatCompletionADMM(Yobs11, indices11, m1,m2, tau=.5, mu=3, rho=(rho*2+1.)/10, tol=.00001, itermax=2000)\n", " print(rho, k)\n", " print(SquaredLossTest(Yobs11,M,indices11), SquaredLossTot(Y11,M), AbsLossTest(Yobs11,M,indices11), AbsLossTot(Y11,M))\n", " print(\"The Nuclear norm is: \", sum(np.linalg.svd(M)[1]))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## Section 2.3 : Fast SVD" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We use the topo from https://jakevdp.github.io/blog/2012/12/19/sparse-svds-in-python/\n", "The first attempt uses ARPACK, a set of Fortran subroutines. PROPACK is not useful to our case because it only suits sparse matrices (in Fortran). The big issue here is that the method \"svds\", called by TruncatedSVD is only suitable for sparse matrices in Python. It's not a Fortran issue. However, it works on regular matrices so we use this version.\n", "Another method in order to compute approximated SVD is to use randomized version." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [], "source": [ "[Y5,Yobs5,indices5]=AbsMatrixCreationA(3000,3000,10,n,0.2) # The difference is enough impressive\n" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([ 2810.71206556, 2924.71410333, 2943.14261837, 2958.38116791,\n", " 2978.56525181, 3020.24047066, 3043.82189401, 3066.0155669 ,\n", " 3099.76474729, 3183.76825908, 0. , 0. ,\n", " 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. ])" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "U, Sigma, VT = svds(Y5, k=20) ; Sigma" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([ 3.18376826e+03, 3.09976475e+03, 3.06601557e+03,\n", " 3.04382189e+03, 3.02024047e+03, 2.97856525e+03,\n", " 2.95838117e+03, 2.94314262e+03, 2.92471410e+03,\n", " 2.81071207e+03, 4.76269673e-12, 4.37701146e-12,\n", " 3.87380262e-12, 3.18430707e-12, 2.82775426e-12,\n", " 2.66483292e-12, 2.49625455e-12, 2.43198955e-12,\n", " 2.42106775e-12, 2.36752491e-12])" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "U, Sigma, V = np.linalg.svd(Y5); Sigma[0:20]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "New Method in order to use Truncated SVD. The new parameter \"l\" is the size of the increment for computing the first k then k+l largest singular values." ] }, { "cell_type": "code", "execution_count": 133, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def Fast_QuantileMatCompletionADMM(Yobs, indices, m1,m2, tau, mu, rho,tol,itermax=1000, init_k=10, increment=5):\n", " n = len(Yobs)\n", " M = np.random.normal(0,1,size=(m1,m2))\n", " N = np.zeros((m1,m2))\n", " U = np.zeros((m1,m2))\n", " \n", " # All the variables to float type:\n", " mu = float(mu) ; rho = float(rho)\n", " \n", " if n != len(indices): print(\"There is a big Problem, check Yobs and indices\")\n", " \n", " # function to evaluate the minimum of:\n", " # tau(y-x)_+ + (1-tau)(x-y)_+ + r/2(x-b)^2\n", " # w.r.t x\n", " \n", " def min_abs(y,b):\n", " x1 = np.minimum(y,b+tau/rho)\n", " x2 = np.maximum(y,b-(1-tau)/rho)\n", " y1 = tau*(y-x1) + rho/2 * (x1-b)**2\n", " y2 = (1-tau)*(x2-y) + rho/2*(x2-b)**2\n", " ind_min = np.where(y1 x:\n", " u,S,v = svds(A,k=K)\n", " K = K + increment\n", " min_SV = S[0] \n", " ## Be careful, reverse order for svds !!!\n", "\n", " j = S <= x\n", " S[j] = 0\n", " j = S > x\n", " S[j] = S[j] - x\n", " l=sum(j)\n", " # Reverse order:\n", " u = u[:,::-1]\n", " S = S[::-1]\n", " v = v[::-1,:]\n", " return(np.dot(u[:,:l],np.dot(np.diag(S[:l]),v[:l,:])))\n", " \n", " def update_N(M,U):\n", " return(FastSoftThresMat(M+U,mu/rho))\n", " \n", " def update_U(M,N,U):\n", " return(U + M -N)\n", " \n", " # Main Loop\n", " \n", " ec = 1000.\n", " k = 0\n", " while(ec>tol and k" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "## Test on larger matrices\n", "[Y5,Yobs5,indices5]=AbsMatrixCreationA(500,500,5,1000*1000*.2,2) \n", "for mu in np.arange(10):\n", " M,N,U,k = QuantileMatCompletionADMM(Yobs5, indices5, m1=500,m2=500, tau=.5, mu=mu*5, rho=2, tol=.00001, itermax=1000)\n", " print(mu, k)\n", " print(SquaredLossTest(Yobs5,M,indices5), SquaredLossTot(Y5,M), AbsLossTest(Yobs5,M,indices5), AbsLossTot(Y5,M))\n", " print(\"The Nuclear norm is: \", sum(np.linalg.svd(M)[1]))\n", " plt.plot(np.linalg.svd(M)[1][0:21], lw=2)\n", " plt.title(\"20 first Singular values of M\")\n", " plt.xlim([-2, 22])" ] }, { "cell_type": "code", "execution_count": 139, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(9, 579)\n", "(2.1895389355841699, 0.29244389801163684, 1.1634994285393603, 0.41010369701054922)\n", "52.8472049236\n" ] } ], "source": [ "startTime = time.time()\n", "M,N,U,k = QuantileMatCompletionADMM(Yobs5, indices5, m1=500,m2=500, tau=.5, mu=15, rho=2, tol=.00001, itermax=1000)\n", "print(mu, k)\n", "print(SquaredLossTest(Yobs5,M,indices5), SquaredLossTot(Y5,M), AbsLossTest(Yobs5,M,indices5), AbsLossTot(Y5,M))\n", "elapsedTime = time.time() - startTime\n", "print(elapsedTime)" ] }, { "cell_type": "code", "execution_count": 143, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(9, 578)\n", "(2.189538996992336, 0.29244395691752806, 1.1634994449034584, 0.41010373511521681)\n", "40.6920361519\n" ] } ], "source": [ "startTime = time.time()\n", "M2,N,U,k = Fast_QuantileMatCompletionADMM(Yobs5, indices5, m1=500,m2=500, tau=.5, mu=15, rho=2,tol=.00001,itermax=1000, init_k=20, increment=10)\n", "print(mu, k)\n", "print(SquaredLossTest(Yobs5,M2,indices5), SquaredLossTot(Y5,M2), AbsLossTest(Yobs5,M2,indices5), AbsLossTot(Y5,M2))\n", "elapsedTime = time.time() - startTime\n", "print(elapsedTime)" ] }, { "cell_type": "code", "execution_count": 142, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([ 1.22650578e-01, 6.62169547e-01, 1.04951415e+00,\n", " 1.40598444e+00, 2.38001292e+00, 3.01595790e+00,\n", " 3.23122104e+00, 3.61151962e+00, 3.92804621e+00,\n", " 5.02955204e+00, 5.26434929e+00, 6.13922810e+00,\n", " 8.55450157e+00, 3.48780807e+02, 3.86811920e+02,\n", " 4.00279189e+02, 4.21216629e+02, 4.42869148e+02,\n", " 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,\n", " 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,\n", " 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,\n", " 0.00000000e+00, 0.00000000e+00, 0.00000000e+00])" ] }, "execution_count": 142, "metadata": {}, "output_type": "execute_result" } ], "source": [ "svds(M2,k=30)[1]" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.11" } }, "nbformat": 4, "nbformat_minor": 0 }