.. _sphx_glr_auto_examples_plot_tv_comparison.py: Total variation regularization ============================== Comparison of solvers with total variation regularization. .. code-block:: python import numpy as np from scipy import linalg import pylab as plt import copt as cp Load an ground truth image and generate the dataset (A, b) as b = A ground_truth + noise , where A is a random matrix. We will now load the ground truth image .. code-block:: python img = cp.datasets.load_img1() n_rows, n_cols = img.shape n_features = n_rows * n_cols np.random.seed(0) n_samples = n_features # set L2 regularization (arbitrarily) to 1/n_samples A = np.random.uniform(-1, 1, size=(n_samples, n_features)) for i in range(A.shape[0]): A[i] /= linalg.norm(A[i]) b = A.dot(img.ravel()) + 1.0 * np.random.randn(n_samples) print(1) def TV(w): img = w.reshape((n_rows, n_cols)) tmp1 = np.abs(np.diff(img, axis=0)) tmp2 = np.abs(np.diff(img, axis=1)) return tmp1.sum() + tmp2.sum() class TotalVariation1DCols: def __init__(self, alpha, n_rows, n_cols): self.alpha = alpha self.n_rows = n_rows self.n_cols = n_cols def __call__(self, x): img = x.reshape((self.n_rows, self.n_cols)) tmp1 = np.abs(np.diff(img, axis=0)) return self.alpha * tmp1.sum() def prox(self, x, step_size): return cp.tv_prox.prox_tv1d_cols( step_size * self.alpha, x, self.n_rows, self.n_cols) class TotalVariation1DRows: def __init__(self, alpha, n_rows, n_cols): self.alpha = alpha self.n_rows = n_rows self.n_cols = n_cols def __call__(self, x): img = x.reshape((self.n_rows, self.n_cols)) tmp2 = np.abs(np.diff(img, axis=1)) return self.alpha * tmp2.sum() def prox(self, x, step_size): return cp.tv_prox.prox_tv1d_rows( step_size * self.alpha, x, self.n_rows, self.n_cols) f, ax = plt.subplots(2, 3, sharey=False) all_alphas = [1e-6, 1e-3, 1e-1] xlim = [0.02, 0.02, 0.1] for i, alpha in enumerate(all_alphas): print(i, alpha) # max_iter = 500 out_tos = cp.minimize_DavisYin( cp.SquaredLoss(A, b), TotalVariation1DCols(alpha, n_rows, n_cols), TotalVariation1DRows(alpha, n_rows, n_cols), np.zeros(n_features), max_iter=max_iter, tol=1e-14, verbose=1, trace=True) tos_nols = cp.minimize_DavisYin( cp.SquaredLoss(A, b), TotalVariation1DCols(alpha, n_rows, n_cols), TotalVariation1DRows(alpha, n_rows, n_cols), np.zeros(n_features), max_iter=max_iter, tol=1e-14, verbose=1, trace=True, backtracking=False) # # trace_gd = Trace(lambda x: obj_fun(x) + alpha * TV(x)) # f = cp.LogisticLoss(A, b, l2_reg) # g = cp.TotalVariation2D(alpha, n_rows, n_cols) # out_gd = cp.minimize_APGD( # f, g, max_iter=max_iter, callback=trace_gd) # ax[0, i].set_title(r'$\lambda=%s$' % alpha) ax[0, i].imshow(out_tos.x.reshape((n_rows, n_cols)), interpolation='nearest', cmap=plt.cm.Blues) ax[0, i].set_xticks(()) ax[0, i].set_yticks(()) # fmin = min(np.min(out_tos.trace_func), np.min(tos_nols.trace_func)) #, np.min(trace_gd.values)) scale = (np.array(out_tos.trace_func) - fmin)[0] plot_tos, = ax[1, i].plot( np.array(out_tos.trace_time), (np.array(out_tos.trace_func) - fmin) / scale, lw=4, marker='o', markevery=10, markersize=10) plot_nols, = ax[1, i].plot( np.array(tos_nols.trace_time), (np.array(tos_nols.trace_func) - fmin) / scale, lw=4, marker='h', markevery=10, markersize=10) # prox_gd, = ax[1, i].plot( # np.array(trace_gd.times), (np.array(trace_gd.values) - fmin) / scale, # lw=4, marker='^', markersize=10, markevery=10, # color=colors[1]) ax[1, i].set_xlabel('Time (in seconds)') ax[1, i].set_yscale('log') #ax[1, i].set_xlim((0, xlim[i])) ax[1, i].grid(True) # plt.gcf().subplots_adjust(bottom=0.15) plt.figlegend( (plot_tos, plot_nols), ('TOS (LS)', 'TOS (no LS)'), ncol=5, scatterpoints=1, loc=(-0.00, -0.0), frameon=False, bbox_to_anchor=[0.05, 0.01]) ax[1, 0].set_ylabel('Objective minus optimum') plt.show() .. image:: /auto_examples/images/sphx_glr_plot_tv_comparison_001.png :align: center .. rst-class:: sphx-glr-script-out Out:: 1 0 1e-06 Iteration 1, prox-grad norm: 2.51993724455, step size: 105.436663076, rho: 1.03 Iteration 2, prox-grad norm: 1.14676486987, step size: 108.599762968, rho: 1.0609 Iteration 3, prox-grad norm: 0.69436510239, step size: 111.857755857, rho: 1.092727 Iteration 4, prox-grad norm: 0.476661757617, step size: 115.213488533, rho: 1.1255088100000001 Iteration 5, prox-grad norm: 0.354094749807, step size: 118.669893189, rho: 1.1592740743 Iteration 6, prox-grad norm: 0.27801924202, step size: 122.229989984, rho: 1.1940522965290001 Iteration 7, prox-grad norm: 0.227066509851, step size: 125.896889684, rho: 1.2298738654248702 Iteration 8, prox-grad norm: 0.19074219033, step size: 129.673796375, rho: 1.2667700813876164 Iteration 9, prox-grad norm: 0.163521919922, step size: 133.564010266, rho: 1.304773183829245 Iteration 10, prox-grad norm: 0.142313446879, step size: 137.570930574, rho: 1.3439163793441222 Iteration 11, prox-grad norm: 0.125281181807, step size: 141.698058491, rho: 1.384233870724446 Iteration 12, prox-grad norm: 0.111269991003, step size: 145.949000246, rho: 1.4257608868461793 Iteration 13, prox-grad norm: 0.0995208138042, step size: 150.327470253, rho: 1.4685337134515648 Iteration 14, prox-grad norm: 0.0895139396385, step size: 154.837294361, rho: 1.512589724855112 Iteration 15, prox-grad norm: 0.0808828391488, step size: 159.482413191, rho: 1.5579674166007653 Iteration 16, prox-grad norm: 0.0733626329098, step size: 164.266885587, rho: 1.6047064390987884 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