├── .DS_Store ├── .ipynb_checkpoints ├── ch02-checkpoint.ipynb ├── ch03-checkpoint.ipynb ├── ch04-checkpoint.ipynb ├── ch05-checkpoint.ipynb ├── ch05_demo-checkpoint.ipynb ├── ch06-checkpoint.ipynb ├── ch07-checkpoint.ipynb ├── ch10-checkpoint.ipynb ├── ch11-checkpoint.ipynb ├── ch12-checkpoint.ipynb ├── ch14-01-checkpoint.ipynb ├── ch14-02-checkpoint.ipynb ├── ch14-03-checkpoint.ipynb ├── ch14-04-checkpoint.ipynb ├── ch14-05-checkpoint.ipynb ├── ch14-06-checkpoint.ipynb ├── ch14-checkpoint.ipynb ├── ch15-01-checkpoint.ipynb ├── ch15-02-checkpoint.ipynb ├── ch15-03-checkpoint.ipynb ├── ch15-04-checkpoint.ipynb ├── ch15-05-checkpoint.ipynb ├── ch15-06-checkpoint.ipynb └── ch15-06-ver2-checkpoint.ipynb ├── A_DCT ├── A_DCT.png ├── A_KSVD_25 ├── A_KSVD_25.png ├── A_KSVD_50 ├── A_KSVD_50.png ├── A_KSVD_75 ├── A_KSVD_75.png ├── A_KSVD_sig10 ├── A_KSVD_sig10.png ├── A_KSVD_sig20 ├── A_KSVD_sig20.png ├── A_dl_25 ├── A_dl_50 ├── A_dl_75 ├── A_high ├── A_l ├── A_low ├── Barbara_K-SVD.png ├── CoherenceIteration.png ├── Grassmannian.png ├── IRLS.png ├── MOD_K-SVD.png ├── NonDiagonalElement.png ├── Pepper_Y_sig20 ├── Pepper_Y_sig20_25 ├── Pepper_Y_sig20_50 ├── Pepper_Y_sig20_75 ├── Peppers_Y_sig20 ├── Peppers_Y_sig20_25 ├── Peppers_Y_sig20_50 ├── Peppers_Y_sig20_75 ├── Peppers_inpaint.png ├── Peppers_inpulse ├── Peppers_inpulse.png ├── Peppers_inpulse_mask ├── Peppers_inpulse_mask_median ├── Peppers_inpulse_recon.png ├── Peppers_inpulse_recon_dct_local_inpainting ├── Peppers_inpulse_recon_dct_local_inpainting_mask ├── Peppers_inpulse_recon_local_inpainting ├── Peppers_inpulse_recon_median ├── Peppers_inpulse_recon_median_mask ├── Peppers_inpulse_recon_sub.png ├── Peppers_inpulse_sign ├── Peppers_recon.png ├── Peppers_recon_DCT_DL.png ├── Peppers_recon_DCT_KSVD.png ├── Peppers_recon_KSVD.png ├── Peppers_recon_Y_sig20_25 ├── Peppers_recon_Y_sig20_25_KSVD ├── Peppers_recon_Y_sig20_25_dl ├── Peppers_recon_Y_sig20_50 ├── Peppers_recon_Y_sig20_50_KSVD ├── Peppers_recon_Y_sig20_50_dl ├── Peppers_recon_Y_sig20_75 ├── Peppers_recon_Y_sig20_75_KSVD ├── Peppers_recon_Y_sig20_75_dl ├── X_1.png ├── X_3.png ├── X_5.png ├── Y_c_global_mca ├── Y_c_local ├── Y_t_global_mca ├── Y_t_local ├── activity.png ├── activity_threshold.png ├── barbara.png ├── barbara_patches.png ├── barbara_sig10 ├── barbara_sig10.png ├── barbara_sig20 ├── barbara_sig20.png ├── c_global ├── c_global.png ├── c_local ├── c_local.png ├── ch02.ipynb ├── ch03.ipynb ├── ch04.ipynb ├── ch05.ipynb ├── ch06.ipynb ├── ch07.ipynb ├── ch10.ipynb ├── ch11.ipynb ├── ch12.ipynb ├── ch14-01.ipynb ├── ch14-02.ipynb ├── ch14-03.ipynb ├── ch14-04.ipynb ├── ch14-05.ipynb ├── ch14-denoise.png ├── ch15-01.ipynb ├── ch15-02.ipynb ├── ch15-03.ipynb ├── ch15-04.ipynb ├── ch15-05.ipynb ├── ch15-06.ipynb ├── darakuron.png ├── dct_dictionary.png ├── dct_shrinkage.png ├── dct_shrinkage_threshold.png ├── deblurring.png ├── dist_S.png ├── dist_S_OMP_LARS.png ├── global_mca.png ├── iteration.png ├── ksvd_dictionary_barbara.png ├── l2_err.png ├── l2_err_OMP_LARS.png ├── lambda.png ├── lena.jpg ├── lena.png ├── lena_200_200.png ├── local_mca.png ├── mask ├── mask_25 ├── mask_50 ├── mask_75 ├── mask_median.png ├── p_l_pca ├── path.png ├── peppers.png ├── recon_bm3d ├── recon_bm3d.png ├── recon_dct_dictionary ├── recon_dct_dictionary.png ├── recon_dct_global_shrinkage_curve ├── recon_dct_global_shrinkage_curve.png ├── recon_dct_shrink ├── recon_dct_shrinkage_curve ├── recon_dct_shrinkage_curve.png ├── recon_ksvd_dictionary ├── recon_ksvd_dictionary.png ├── recon_nlm ├── recon_nlm.png ├── recon_ws ├── text.jpg ├── wavelet.png ├── wavelet_shrinkage.png └── wavelet_shrinkage_threshold.png /.DS_Store: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/kibo35/sparse-modeling/56f4263d44224dd08699c7bc854028a53b99871e/.DS_Store -------------------------------------------------------------------------------- /.ipynb_checkpoints/ch04-checkpoint.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "第4章 追跡アルゴリズムの性能保証\n", 8 | "==" 9 | ] 10 | }, 11 | { 12 | "cell_type": "markdown", 13 | "metadata": {}, 14 | "source": [ 15 | "定理4.1（OMPの最適解保証：二つの直交行列の場合）\n", 16 | "--\n", 17 | "$\\mathbf{\\Psi}$,$\\mathbf{\\Phi}$を$n×n$の直交行列とする．線形連立方程式$\\mathbf{Ax}=[\\mathbf{\\Psi},\\mathbf{\\Phi}]\\mathbf{x}=\\mathbf{b}$に対して，もし，解$\\mathbf{x}$が存在して、最初の$n$個の要素の中に$k_p$個の非ゼロ要素があり，\n", 18 | "$$max(k_p,k_q)<\\frac{1}{2\\mu(\\mathbf{A})}$$\n", 19 | "が成り立つならば，しきい値パラメータ$\\epsilon_0=0$で実行されるOMPは$k_0=k_p+K_q$回の反復で最適解を与える．" 20 | ] 21 | }, 22 | { 23 | "cell_type": "markdown", 24 | "metadata": {}, 25 | "source": [ 26 | "定理4.2（基底追跡の最適解保証：二つの直交行列の場合）\n", 27 | "--\n", 28 | "$\\mathbf{\\Psi}$,$\\mathbf{\\Phi}$を$n×n$の直交行列とする．線形連立方程式$\\mathbf{Ax}=[\\mathbf{\\Psi},\\mathbf{\\Phi}]\\mathbf{x}=\\mathbf{b}$に対して，もし，解$\\mathbf{x}$が存在して、最初の$n$個の要素の中に$k_p$個の非ゼロ要素があり，\n", 29 | "$$2\\mu(\\mathbf{A})^2k_pk_q+\\mu(\\mathbf{A})k_p-1<0$$\n", 30 | "が成り立つならば，得られた解は$(P_1)$の一意解であり，${P_0}$の一意解でもある．" 31 | ] 32 | }, 33 | { 34 | "cell_type": "markdown", 35 | "metadata": {}, 36 | "source": [ 37 | "線形計画問題(LP)\n", 38 | "--\n", 39 | "\n", 40 | "BP（基底追跡）の解$\\mathbf{x}$より$L_0$ノルムが大きくて$L_1$ノルムが小さい解$\\mathbf{y}$の集合\n", 41 | "\n", 42 | "$C = \\{\\mathbf{y}\\,|\\,\n", 43 | "\\mathbf{y}\\neq \\mathbf{x},\\,\n", 44 | "\\|\\mathbf{y}\\|_1\\leq\\|\\mathbf{x}\\|_1,\\,\n", 45 | "\\|\\mathbf{y}\\|_0\\geq\\|\\mathbf{x}\\|_0,\\,\n", 46 | "\\mathbf{A}(\\mathbf{y}-\\mathbf{x})=\\mathbf{0}\\}\n", 47 | "$\n", 48 | "\n", 49 | "が与えられた条件\n", 50 | "\n", 51 | "$2\\mu(\\mathbf{A})^2k_pk_q+\\mu(\\mathbf{A})k_p-1<0$
\n", 52 | "\n", 53 | "もしくはもっと解釈が容易な条件\n", 54 | "\n", 55 | "$\\|x\\|_0=k_p+k_q<\\frac{\\sqrt{2}-0.5}{\\mu(\\mathbf{A})}$\n", 56 | "\n", 57 | "の元で空となることを証明する" 58 | ] 59 | }, 60 | { 61 | "cell_type": "code", 62 | "execution_count": 55, 63 | "metadata": { 64 | "collapsed": false 65 | }, 66 | "outputs": [ 67 | { 68 | "data": { 69 | "text/plain": [ 70 | " fun: 50.0\n", 71 | " message: 'Optimization failed. Unable to find a feasible starting point.'\n", 72 | " nit: 100\n", 73 | " status: 2\n", 74 | " success: False\n", 75 | " x: nan" 76 | ] 77 | }, 78 | "execution_count": 55, 79 | "metadata": {}, 80 | "output_type": "execute_result" 81 | } 82 | ], 83 | "source": [ 84 | "from scipy.optimize import linprog\n", 85 | "\n", 86 | "# パラメータ\n", 87 | "n, kp, kq, mu = 50, 9, 7, 0.1\n", 88 | "\n", 89 | "one_p = [1.] * kp + [0.] * (n - kp)\n", 90 | "one_q = [1.] * kq + [0.] * (n - kq)\n", 91 | "C = np.array(one_p + one_q)\n", 92 | "A_ub = np.diag([1. - mu] * n + [mu - 1.] * n)\n", 93 | "b_ub = np.zeros(2 * n)\n", 94 | "A_eq = np.eye(2 * n)\n", 95 | "b_eq = np.ones(2 * n)\n", 96 | "\n", 97 | "# 最大化したいので負をとる\n", 98 | "C *= -1\n", 99 | "A_ub *= -1\n", 100 | "b_ub *= -1\n", 101 | "A_eq *= -1\n", 102 | "b_eq *= -1\n", 103 | "\n", 104 | "linprog(C, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq)" 105 | ] 106 | }, 107 | { 108 | "cell_type": "markdown", 109 | "metadata": {}, 110 | "source": [ 111 | "nanだと…解集合が空であることを示したいからこれでいいのか…" 112 | ] 113 | }, 114 | { 115 | "cell_type": "markdown", 116 | "metadata": {}, 117 | "source": [ 118 | "定理4.3（OMPの最適解保証）\n", 119 | "--\n", 120 | "線形連立方程式$\\mathbf{Ax}=\\mathbf{b}$（$n\n", 132 | "（その非ゼロ要素の最小値は$|x_{min}|$，最大値は$|x_{max}|$とする）\n", 133 | "$$\\|\\mathbf{x}\\|_0<\\frac{1}{2}\\left(1+\\frac{1}{\\mu(\\mathbf{A})}\\frac{|x_{min}|}{|x_{max}|}\\right)$$\n", 134 | "を満たせば，しきい値パラメータ$\\epsilon_0=0$で実行されるしきい値アルゴリズムは最適解を与える．" 135 | ] 136 | }, 137 | { 138 | "cell_type": "markdown", 139 | "metadata": {}, 140 | "source": [ 141 | "定理4.5（基底追跡の最適解保証）\n", 142 | "--\n", 143 | "線形連立方程式$\\mathbf{Ax}=\\mathbf{b}$（$n" 251 | ] 252 | }, 253 | "execution_count": 57, 254 | "metadata": {}, 255 | "output_type": "execute_result" 256 | }, 257 | { 258 | "data": { 259 | "image/png": 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a3dss9Q8ot/LkapPr169TvXp1jh49au/wFxoaSr169Th+/Libo7vt0qVLNG7cmJ9//pk8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\nR60/pio96dzZembmxIng5YW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260 | "text/plain": [ 261 | "" 262 | ] 263 | }, 264 | "metadata": {}, 265 | "output_type": "display_data" 266 | } 267 | ], 268 | "source": [ 269 | "x = np.linspace(0, 4, 100)\n", 270 | "\n", 271 | "y0 = (-3*x+10)/2\n", 272 | "y1 = -2*x+6\n", 273 | "y2 = -x+4\n", 274 | "y3 = np.ones_like(x)\n", 275 | "\n", 276 | "y4 = np.minimum(y1, y2)\n", 277 | "\n", 278 | "plt.figure()\n", 279 | "plt.plot(x, y0, \"--\", label=\"objective function\")\n", 280 | "plt.plot(x, y1, label=\"2x+y<=6\")\n", 281 | "plt.plot(x, y2, label=\"x+y<=4\")\n", 282 | "plt.plot(x, y3, label=\"y>=1\")\n", 283 | "plt.plot(res.x[0], res.x[1], \"o\")\n", 284 | "plt.fill_between(x, y3, y4, where=y4>y3, facecolor='yellow', alpha=0.3)\n", 285 | "plt.ylim(0, 5)\n", 286 | "plt.legend(loc=0)" 287 | ] 288 | }, 289 | { 290 | "cell_type": "markdown", 291 | "metadata": {}, 292 | "source": [ 293 | "PuLP で解くと以下のようになります。" 294 | ] 295 | }, 296 | { 297 | "cell_type": "code", 298 | "execution_count": 58, 299 | "metadata": { 300 | "collapsed": false 301 | }, 302 | "outputs": [ 303 | { 304 | "data": { 305 | "text/plain": [ 306 | "[2.0, 2.0]" 307 | ] 308 | }, 309 | "execution_count": 58, 310 | "metadata": {}, 311 | "output_type": "execute_result" 312 | } 313 | ], 314 | "source": [ 315 | "import pulp\n", 316 | "\n", 317 | "x = pulp.LpVariable(\"x\")\n", 318 | "y = pulp.LpVariable(\"y\")\n", 319 | "problem = pulp.LpProblem(\"A simple maximization objective\", pulp.LpMaximize)\n", 320 | "problem += 3*x + 2*y, \"The objective function\"\n", 321 | "problem += 2*x + y <= 6, \"1st constraint\"\n", 322 | "problem += x + y <= 4, \"2nd constraint\"\n", 323 | "problem += y >= 1, \"3rd constraint\"\n", 324 | "problem.solve()\n", 325 | "\n", 326 | "[x.value(), y.value()]" 327 | ] 328 | }, 329 | { 330 | "cell_type": "code", 331 | "execution_count": null, 332 | "metadata": { 333 | "collapsed": true 334 | }, 335 | "outputs": [], 336 | "source": [] 337 | } 338 | ], 339 | "metadata": { 340 | "kernelspec": { 341 | "display_name": "Python 3", 342 | "language": "python", 343 | "name": "python3" 344 | }, 345 | "language_info": { 346 | "codemirror_mode": { 347 | "name": "ipython", 348 | "version": 3 349 | }, 350 | "file_extension": ".py", 351 | "mimetype": "text/x-python", 352 | "name": "python", 353 | "nbconvert_exporter": "python", 354 | "pygments_lexer": "ipython3", 355 | "version": "3.5.2" 356 | } 357 | }, 358 | "nbformat": 4, 359 | "nbformat_minor": 0 360 | } 361 | -------------------------------------------------------------------------------- /A_DCT: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/kibo35/sparse-modeling/56f4263d44224dd08699c7bc854028a53b99871e/A_DCT -------------------------------------------------------------------------------- /A_DCT.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/kibo35/sparse-modeling/56f4263d44224dd08699c7bc854028a53b99871e/A_DCT.png 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-------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "第4章 追跡アルゴリズムの性能保証\n", 8 | "==" 9 | ] 10 | }, 11 | { 12 | "cell_type": "markdown", 13 | "metadata": {}, 14 | "source": [ 15 | "定理4.1（OMPの最適解保証：二つの直交行列の場合）\n", 16 | "--\n", 17 | "$\\mathbf{\\Psi}$,$\\mathbf{\\Phi}$を$n×n$の直交行列とする．線形連立方程式$\\mathbf{Ax}=[\\mathbf{\\Psi},\\mathbf{\\Phi}]\\mathbf{x}=\\mathbf{b}$に対して，もし，解$\\mathbf{x}$が存在して、最初の$n$個の要素の中に$k_p$個の非ゼロ要素があり，\n", 18 | "$$max(k_p,k_q)<\\frac{1}{2\\mu(\\mathbf{A})}$$\n", 19 | "が成り立つならば，しきい値パラメータ$\\epsilon_0=0$で実行されるOMPは$k_0=k_p+K_q$回の反復で最適解を与える．" 20 | ] 21 | }, 22 | { 23 | "cell_type": "markdown", 24 | "metadata": {}, 25 | "source": [ 26 | "定理4.2（基底追跡の最適解保証：二つの直交行列の場合）\n", 27 | "--\n", 28 | "$\\mathbf{\\Psi}$,$\\mathbf{\\Phi}$を$n×n$の直交行列とする．線形連立方程式$\\mathbf{Ax}=[\\mathbf{\\Psi},\\mathbf{\\Phi}]\\mathbf{x}=\\mathbf{b}$に対して，もし，解$\\mathbf{x}$が存在して、最初の$n$個の要素の中に$k_p$個の非ゼロ要素があり，\n", 29 | "$$2\\mu(\\mathbf{A})^2k_pk_q+\\mu(\\mathbf{A})k_p-1<0$$\n", 30 | "が成り立つならば，得られた解は$(P_1)$の一意解であり，${P_0}$の一意解でもある．" 31 | ] 32 | }, 33 | { 34 | "cell_type": "markdown", 35 | "metadata": {}, 36 | "source": [ 37 | "線形計画問題(LP)\n", 38 | "--\n", 39 | "\n", 40 | "BP（基底追跡）の解$\\mathbf{x}$より$L_0$ノルムが大きくて$L_1$ノルムが小さい解$\\mathbf{y}$の集合\n", 41 | "\n", 42 | "$C = \\{\\mathbf{y}\\,|\\,\n", 43 | "\\mathbf{y}\\neq \\mathbf{x},\\,\n", 44 | "\\|\\mathbf{y}\\|_1\\leq\\|\\mathbf{x}\\|_1,\\,\n", 45 | "\\|\\mathbf{y}\\|_0\\geq\\|\\mathbf{x}\\|_0,\\,\n", 46 | "\\mathbf{A}(\\mathbf{y}-\\mathbf{x})=\\mathbf{0}\\}\n", 47 | "$\n", 48 | "\n", 49 | "が与えられた条件\n", 50 | "\n", 51 | "$2\\mu(\\mathbf{A})^2k_pk_q+\\mu(\\mathbf{A})k_p-1<0$
\n", 52 | "\n", 53 | "もしくはもっと解釈が容易な条件\n", 54 | "\n", 55 | "$\\|x\\|_0=k_p+k_q<\\frac{\\sqrt{2}-0.5}{\\mu(\\mathbf{A})}$\n", 56 | "\n", 57 | "の元で空となることを証明する" 58 | ] 59 | }, 60 | { 61 | "cell_type": "code", 62 | "execution_count": 55, 63 | "metadata": { 64 | "collapsed": false 65 | }, 66 | "outputs": [ 67 | { 68 | "data": { 69 | "text/plain": [ 70 | " fun: 50.0\n", 71 | " message: 'Optimization failed. Unable to find a feasible starting point.'\n", 72 | " nit: 100\n", 73 | " status: 2\n", 74 | " success: False\n", 75 | " x: nan" 76 | ] 77 | }, 78 | "execution_count": 55, 79 | "metadata": {}, 80 | "output_type": "execute_result" 81 | } 82 | ], 83 | "source": [ 84 | "from scipy.optimize import linprog\n", 85 | "\n", 86 | "# パラメータ\n", 87 | "n, kp, kq, mu = 50, 9, 7, 0.1\n", 88 | "\n", 89 | "one_p = [1.] * kp + [0.] * (n - kp)\n", 90 | "one_q = [1.] * kq + [0.] * (n - kq)\n", 91 | "C = np.array(one_p + one_q)\n", 92 | "A_ub = np.diag([1. - mu] * n + [mu - 1.] * n)\n", 93 | "b_ub = np.zeros(2 * n)\n", 94 | "A_eq = np.eye(2 * n)\n", 95 | "b_eq = np.ones(2 * n)\n", 96 | "\n", 97 | "# 最大化したいので負をとる\n", 98 | "C *= -1\n", 99 | "A_ub *= -1\n", 100 | "b_ub *= -1\n", 101 | "A_eq *= -1\n", 102 | "b_eq *= -1\n", 103 | "\n", 104 | "linprog(C, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq)" 105 | ] 106 | }, 107 | { 108 | "cell_type": "markdown", 109 | "metadata": {}, 110 | "source": [ 111 | "nanだと…解集合が空であることを示したいからこれでいいのか…" 112 | ] 113 | }, 114 | { 115 | "cell_type": "markdown", 116 | "metadata": {}, 117 | "source": [ 118 | "定理4.3（OMPの最適解保証）\n", 119 | "--\n", 120 | "線形連立方程式$\\mathbf{Ax}=\\mathbf{b}$（$n\n", 132 | "（その非ゼロ要素の最小値は$|x_{min}|$，最大値は$|x_{max}|$とする）\n", 133 | "$$\\|\\mathbf{x}\\|_0<\\frac{1}{2}\\left(1+\\frac{1}{\\mu(\\mathbf{A})}\\frac{|x_{min}|}{|x_{max}|}\\right)$$\n", 134 | "を満たせば，しきい値パラメータ$\\epsilon_0=0$で実行されるしきい値アルゴリズムは最適解を与える．" 135 | ] 136 | }, 137 | { 138 | "cell_type": "markdown", 139 | "metadata": {}, 140 | "source": [ 141 | "定理4.5（基底追跡の最適解保証）\n", 142 | "--\n", 143 | "線形連立方程式$\\mathbf{Ax}=\\mathbf{b}$（$n" 251 | ] 252 | }, 253 | "execution_count": 57, 254 | "metadata": {}, 255 | "output_type": "execute_result" 256 | }, 257 | { 258 | "data": { 259 | "image/png": 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a3dss9Q8ot/LkapPr169TvXp1jh49au/wFxoaSr169Th+/Libo7vt0qVLNG7cmJ9//pk8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\nR60/pio96dzZembmxIng5YW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260 | "text/plain": [ 261 | "" 262 | ] 263 | }, 264 | "metadata": {}, 265 | "output_type": "display_data" 266 | } 267 | ], 268 | "source": [ 269 | "x = np.linspace(0, 4, 100)\n", 270 | "\n", 271 | "y0 = (-3*x+10)/2\n", 272 | "y1 = -2*x+6\n", 273 | "y2 = -x+4\n", 274 | "y3 = np.ones_like(x)\n", 275 | "\n", 276 | "y4 = np.minimum(y1, y2)\n", 277 | "\n", 278 | "plt.figure()\n", 279 | "plt.plot(x, y0, \"--\", label=\"objective function\")\n", 280 | "plt.plot(x, y1, label=\"2x+y<=6\")\n", 281 | "plt.plot(x, y2, label=\"x+y<=4\")\n", 282 | "plt.plot(x, y3, label=\"y>=1\")\n", 283 | "plt.plot(res.x[0], res.x[1], \"o\")\n", 284 | "plt.fill_between(x, y3, y4, where=y4>y3, facecolor='yellow', alpha=0.3)\n", 285 | "plt.ylim(0, 5)\n", 286 | "plt.legend(loc=0)" 287 | ] 288 | }, 289 | { 290 | "cell_type": "markdown", 291 | "metadata": {}, 292 | "source": [ 293 | "PuLP で解くと以下のようになります。" 294 | ] 295 | }, 296 | { 297 | "cell_type": "code", 298 | "execution_count": 58, 299 | "metadata": { 300 | "collapsed": false 301 | }, 302 | "outputs": [ 303 | { 304 | "data": { 305 | "text/plain": [ 306 | "[2.0, 2.0]" 307 | ] 308 | }, 309 | "execution_count": 58, 310 | "metadata": {}, 311 | "output_type": "execute_result" 312 | } 313 | ], 314 | "source": [ 315 | "import pulp\n", 316 | "\n", 317 | "x = pulp.LpVariable(\"x\")\n", 318 | "y = pulp.LpVariable(\"y\")\n", 319 | "problem = pulp.LpProblem(\"A simple maximization objective\", pulp.LpMaximize)\n", 320 | "problem += 3*x + 2*y, \"The objective function\"\n", 321 | "problem += 2*x + y <= 6, \"1st constraint\"\n", 322 | "problem += x + y <= 4, \"2nd constraint\"\n", 323 | "problem += y >= 1, \"3rd constraint\"\n", 324 | "problem.solve()\n", 325 | "\n", 326 | "[x.value(), y.value()]" 327 | ] 328 | }, 329 | { 330 | "cell_type": "code", 331 | "execution_count": null, 332 | "metadata": { 333 | "collapsed": true 334 | }, 335 | "outputs": [], 336 | "source": [] 337 | } 338 | ], 339 | "metadata": { 340 | "kernelspec": { 341 | "display_name": "Python 3", 342 | "language": "python", 343 | "name": "python3" 344 | }, 345 | "language_info": { 346 | "codemirror_mode": { 347 | "name": "ipython", 348 | "version": 3 349 | }, 350 | "file_extension": ".py", 351 | "mimetype": "text/x-python", 352 | "name": "python", 353 | "nbconvert_exporter": "python", 354 | "pygments_lexer": "ipython3", 355 | "version": "3.5.2" 356 | } 357 | }, 358 | "nbformat": 4, 359 | "nbformat_minor": 0 360 | } 361 | -------------------------------------------------------------------------------- /ch14-denoise.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/kibo35/sparse-modeling/56f4263d44224dd08699c7bc854028a53b99871e/ch14-denoise.png -------------------------------------------------------------------------------- /darakuron.png: -------------------------------------------------------------------------------- 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