Total variation regularization

Comparison of solvers with total variation regularization.

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

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()

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
Iteration 17, prox-grad norm: 0.0667568838868, step size: 169.194892155, rho: 1.652847632271752
Iteration 18, prox-grad norm: 0.0609176405375, step size: 174.270738919, rho: 1.7024330612399046
Iteration 19, prox-grad norm: 0.0557294613965, step size: 179.498861087, rho: 1.7535060530771018
Iteration 20, prox-grad norm: 0.0511009897396, step size: 184.88382692, rho: 1.806111234669415
Iteration 21, prox-grad norm: 0.0469594480154, step size: 190.430341727, rho: 1.8602945717094976
Iteration 22, prox-grad norm: 0.0432441451979, step size: 196.143251979, rho: 1.9161034088607827
Iteration 23, prox-grad norm: 0.0399042245044, step size: 202.027549538, rho: 1.9735865111266062
Iteration 24, prox-grad norm: 0.0368966699937, step size: 208.088376025, rho: 2.0327941064604045
Iteration 25, prox-grad norm: 0.034184438409, step size: 214.331027305, rho: 2.0937779296542165
Iteration 26, prox-grad norm: 0.0317347818497, step size: 220.760958125, rho: 2.1565912675438432
Iteration 27, prox-grad norm: 0.0295195468138, step size: 227.383786868, rho: 2.2212890055701586
Iteration 28, prox-grad norm: 0.0275138166652, step size: 234.205300474, rho: 2.2879276757372633
Iteration 29, prox-grad norm: 0.0256959095123, step size: 241.231459489, rho: 2.3565655060093813
Iteration 30, prox-grad norm: 0.0240451483243, step size: 248.468403273, rho: 2.4272624711896627
Iteration 31, prox-grad norm: 0.0225444089249, step size: 255.922455371, rho: 2.5000803453253524
Iteration 32, prox-grad norm: 0.0211770820294, step size: 263.600129033, rho: 2.5750827556851132
Iteration 33, prox-grad norm: 0.0199298216049, step size: 271.508132904, rho: 2.6523352383556666
Iteration 34, prox-grad norm: 0.0187894370567, step size: 279.653376891, rho: 2.7319052955063365
Iteration 35, prox-grad norm: 0.0177443854709, step size: 288.042978197, rho: 2.8138624543715265
Iteration 36, prox-grad norm: 0.0167853010171, step size: 296.684267543, rho: 2.8982783280026725
Iteration 37, prox-grad norm: 0.0159024101542, step size: 305.58479557, rho: 2.9852266778427525
Iteration 38, prox-grad norm: 0.0150877202071, step size: 314.752339437, rho: 3.074783478178035
Iteration 39, prox-grad norm: 0.0143341139367, step size: 324.19490962, rho: 3.1670269825233763
Iteration 40, prox-grad norm: 0.0136365668988, step size: 333.920756908, rho: 3.2620377919990777
Iteration 41, prox-grad norm: 0.0129924028256, step size: 343.938379616, rho: 3.35989892575905
Iteration 42, prox-grad norm: 0.0124194490992, step size: 354.256531004, rho: 3.4606958935318217
Iteration 43, prox-grad norm: 0.0120411563419, step size: 364.884226934, rho: 3.5645167703377765
Iteration 44, prox-grad norm: 0.0126065672103, step size: 375.830753742, rho: 3.67145227344791
Iteration 45, prox-grad norm: 0.0182895438337, step size: 150.332301497, rho: 1.4685809093791642
Iteration 46, prox-grad norm: 0.0123103104219, step size: 154.842270542, rho: 1.5126383366605392
Iteration 47, prox-grad norm: 0.0107759891606, step size: 159.487538658, rho: 1.5580174867603553
Iteration 48, prox-grad norm: 0.0103130281816, step size: 164.272164818, rho: 1.604758011363166
Iteration 49, prox-grad norm: 0.0100649469774, step size: 169.200329762, rho: 1.652900751704061
Iteration 50, prox-grad norm: 0.00986562557951, step size: 174.276339655, rho: 1.702487774255183
Iteration 51, prox-grad norm: 0.00967950502716, step size: 179.504629845, rho: 1.7535624074828384
Iteration 52, prox-grad norm: 0.00949796876375, step size: 184.88976874, rho: 1.8061692797073237
Iteration 53, prox-grad norm: 0.00931850330306, step size: 190.436461802, rho: 1.8603543580985433
Iteration 54, prox-grad norm: 0.00914066779949, step size: 196.149555656, rho: 1.9161649888414998
Iteration 55, prox-grad norm: 0.00896402809341, step size: 202.034042326, rho: 1.9736499385067447
Iteration 56, prox-grad norm: 0.00878861900402, step size: 208.095063596, rho: 2.0328594366619472
Iteration 57, prox-grad norm: 0.00861431852363, step size: 214.337915504, rho: 2.0938452197618056
Iteration 58, prox-grad norm: 0.00844142119154, step size: 220.768052969, rho: 2.15666057635466
Iteration 59, prox-grad norm: 0.00826959411751, step size: 227.391094558, rho: 2.2213603936452997
Iteration 60, prox-grad norm: 0.00809899563431, step size: 234.212827395, rho: 2.2880012054546586
Iteration 61, prox-grad norm: 0.00793025066113, step size: 241.239212217, rho: 2.3566412416182985
Iteration 62, prox-grad norm: 0.00776275450823, step size: 248.476388583, rho: 2.4273404788668476
Iteration 63, prox-grad norm: 0.00759756578143, step size: 255.930680241, rho: 2.500160693232853
Iteration 64, prox-grad norm: 0.0074356881997, step size: 263.608600648, rho: 2.575165514029839
Iteration 65, prox-grad norm: 0.00728044663353, step size: 271.516858667, rho: 2.652420479450734
Iteration 66, prox-grad norm: 0.00714005760187, step size: 279.662364427, rho: 2.731993093834256
Iteration 67, prox-grad norm: 0.00704324117102, step size: 288.05223536, rho: 2.813952886649284
Iteration 68, prox-grad norm: 0.00708279548269, step size: 296.693802421, rho: 2.8983714732487624
Iteration 69, prox-grad norm: 0.00757594422306, step size: 305.594616493, rho: 2.9853226174462253
Iteration 70, prox-grad norm: 0.00976968449632, step size: 122.237846597, rho: 1.1941290469784902
Iteration 71, prox-grad norm: 0.00654454924495, step size: 125.904981995, rho: 1.229952918387845
Iteration 72, prox-grad norm: 0.00635390842992, step size: 129.682131455, rho: 1.2668515059394805
Iteration 73, prox-grad norm: 0.00628773371161, step size: 133.572595399, rho: 1.304857051117665
Iteration 74, prox-grad norm: 0.00622528732137, step size: 137.579773261, rho: 1.344002762651195
Iteration 75, prox-grad norm: 0.00616113739883, step size: 141.707166459, rho: 1.3843228455307308
Iteration 76, prox-grad norm: 0.00609720199489, step size: 145.958381452, rho: 1.4258525308966528
Iteration 77, prox-grad norm: 0.00603268856003, step size: 150.337132896, rho: 1.4686281068235523
Iteration 78, prox-grad norm: 0.00596712004509, step size: 154.847246883, rho: 1.512686950028259
Iteration 79, prox-grad norm: 0.00590114181559, step size: 159.492664289, rho: 1.5580675585291068
Iteration 80, prox-grad norm: 0.00583445352769, step size: 164.277444218, rho: 1.6048095852849802
Iteration 81, prox-grad norm: 0.00576707430502, step size: 169.205767545, rho: 1.6529538728435296
Iteration 82, prox-grad norm: 0.00569856483778, step size: 174.281940571, rho: 1.7025424890288354
Iteration 83, prox-grad norm: 0.00562987041268, step size: 179.510398788, rho: 1.7536187636997005
Iteration 84, prox-grad norm: 0.00556054987688, step size: 184.895710752, rho: 1.8062273266106916
Iteration 85, prox-grad norm: 0.00549019994804, step size: 190.442582074, rho: 1.8604141464090125
Iteration 86, prox-grad norm: 0.00541970315499, step size: 196.155859536, rho: 1.916226570801283
Iteration 87, prox-grad norm: 0.00534826013062, step size: 202.040535322, rho: 1.9737133679253214
Iteration 88, prox-grad norm: 0.0052766186857, step size: 208.101751382, rho: 2.032924768963081
Iteration 89, prox-grad norm: 0.00520435034523, step size: 214.344803924, rho: 2.0939125120319737
Iteration 90, prox-grad norm: 0.00513182952015, step size: 220.775148041, rho: 2.156729887392933
Iteration 91, prox-grad norm: 0.00505853668794, step size: 227.398402483, rho: 2.2214317840147206
Iteration 92, prox-grad norm: 0.00498497120021, step size: 234.220354557, rho: 2.288074737535162
Iteration 93, prox-grad norm: 0.0049109092608, step size: 241.246965194, rho: 2.356716979661217
Iteration 94, prox-grad norm: 0.00483670533052, step size: 248.48437415, rho: 2.4274184890510537
Iteration 95, prox-grad norm: 0.00476214989222, step size: 255.938905374, rho: 2.5002410437225855
Iteration 96, prox-grad norm: 0.00468738818494, step size: 263.617072535, rho: 2.575248275034263
Iteration 97, prox-grad norm: 0.00461254875299, step size: 271.525584711, rho: 2.652505723285291
Iteration 98, prox-grad norm: 0.00453751478972, step size: 279.671352253, rho: 2.7320808949838495
Iteration 99, prox-grad norm: 0.00446232425782, step size: 288.06149282, rho: 2.814043321833365
Iteration 100, prox-grad norm: 0.0043867065474, step size: 296.703337605, rho: 2.8984646214883663
Iteration 101, prox-grad norm: 0.00431076321422, step size: 305.604437733, rho: 2.9854185601330174
Iteration 102, prox-grad norm: 0.00423497386509, step size: 314.772570865, rho: 3.074981116937008
Iteration 103, prox-grad norm: 0.00415967956495, step size: 324.215747991, rho: 3.167230550445118
Iteration 104, prox-grad norm: 0.00408420248343, step size: 333.942220431, rho: 3.2622474669584713
Iteration 105, prox-grad norm: 0.0040106817122, step size: 343.960487044, rho: 3.3601148909672256
Iteration 106, prox-grad norm: 0.00394462698886, step size: 354.279301655, rho: 3.4609183376962425
Iteration 107, prox-grad norm: 0.00391947769842, step size: 364.907680705, rho: 3.56474588782713
Iteration 108, prox-grad norm: 0.00411923032428, step size: 375.854911126, rho: 3.671688264461944
Iteration 109, prox-grad norm: 0.00561795960367, step size: 150.34196445, rho: 1.4686753057847777
Iteration 110, prox-grad norm: 0.00405270773953, step size: 154.852223384, rho: 1.512735564958321
Iteration 111, prox-grad norm: 0.00372141918617, step size: 159.497790085, rho: 1.5581176319070706
Iteration 112, prox-grad norm: 0.00363569372989, step size: 164.282723788, rho: 1.6048611608642829
Iteration 113, prox-grad norm: 0.00359330658763, step size: 169.211205502, rho: 1.6530069956902114
Iteration 114, prox-grad norm: 0.00355945009122, step size: 174.287541667, rho: 1.7025972055609178
Iteration 115, prox-grad norm: 0.00352729397133, step size: 179.516167917, rho: 1.7536751217277453
Iteration 116, prox-grad norm: 0.00349480205768, step size: 184.901652954, rho: 1.8062853753795778
Iteration 117, prox-grad norm: 0.00346253673415, step size: 190.448702543, rho: 1.8604739366409653
Iteration 118, prox-grad norm: 0.00342979373966, step size: 196.162163619, rho: 1.9162881547401942
Iteration 119, prox-grad norm: 0.00339662477013, step size: 202.047028528, rho: 1.9737767993824
Iteration 120, prox-grad norm: 0.00336290288312, step size: 208.108439383, rho: 2.032990103363872
Iteration 121, prox-grad norm: 0.00332896431374, step size: 214.351692565, rho: 2.0939798064647883
Iteration 122, prox-grad norm: 0.0032944146045, step size: 220.782243342, rho: 2.156799200658732
Iteration 123, prox-grad norm: 0.00325969428407, step size: 227.405710642, rho: 2.2215031766784943
Iteration 124, prox-grad norm: 0.0032245795508, step size: 234.227881961, rho: 2.2881482719788493
Iteration 125, prox-grad norm: 0.00318886958955, step size: 241.25471842, rho: 2.356792720138215
Iteration 126, prox-grad norm: 0.00315287194119, step size: 248.492359973, rho: 2.4274965017423615
Iteration 127, prox-grad norm: 0.00311670461992, step size: 255.947130772, rho: 2.5003213967946323
Iteration 128, prox-grad norm: 0.00307995427654, step size: 263.625544695, rho: 2.5753310386984714
Iteration 129, prox-grad norm: 0.00304339541097, step size: 271.534311036, rho: 2.6525909698594257
Iteration 130, prox-grad norm: 0.00300737660938, step size: 279.680340367, rho: 2.7321686989552085
Iteration 131, prox-grad norm: 0.00297320863722, step size: 288.070750578, rho: 2.8141337599238647
Iteration 132, prox-grad norm: 0.00294598544401, step size: 296.712873095, rho: 2.8985577727215808
Iteration 133, prox-grad norm: 0.00294265136419, step size: 305.614259288, rho: 2.9855145059032284
Iteration 134, prox-grad norm: 0.00302731512419, step size: 314.782687067, rho: 3.0750799410803253
Iteration 135, prox-grad norm: 0.00342930775602, step size: 324.226167679, rho: 3.1673323393127353
Iteration 136, prox-grad norm: 0.00498592392374, step size: 129.690467072, rho: 1.2669329357250942
Iteration 137, prox-grad norm: 0.00292622614144, step size: 133.581181084, rho: 1.304940923796847
Iteration 138, prox-grad norm: 0.0027541521929, step size: 137.588616516, rho: 1.3440891515107525
Iteration 139, prox-grad norm: 0.00272875132062, step size: 141.716275012, rho: 1.3844118260560752
Iteration 140, prox-grad norm: 0.00271167185503, step size: 145.967763262, rho: 1.4259441808377575
Iteration 141, prox-grad norm: 0.00269515854768, step size: 150.34679616, rho: 1.4687225062628904
Iteration 142, prox-grad norm: 0.00267839801817, step size: 154.857200045, rho: 1.5127841814507772
Iteration 143, prox-grad norm: 0.00266122666073, step size: 159.502916046, rho: 1.5581677068943005
Iteration 144, prox-grad norm: 0.00264364164493, step size: 164.288003527, rho: 1.6049127381011297
Iteration 145, prox-grad norm: 0.00262576719553, step size: 169.216643633, rho: 1.6530601202441635
Iteration 146, prox-grad norm: 0.00260776076575, step size: 174.293142942, rho: 1.7026519238514883
Iteration 147, prox-grad norm: 0.0025894388375, step size: 179.521937231, rho: 1.753731481567033
Iteration 148, prox-grad norm: 0.00257080126438, step size: 184.907595347, rho: 1.806343426014044
Iteration 149, prox-grad norm: 0.00255156309002, step size: 190.454823208, rho: 1.8605337287944654
Iteration 150, prox-grad norm: 0.00253206329781, step size: 196.168467904, rho: 1.9163497406582994
Iteration 151, prox-grad norm: 0.00251232638725, step size: 202.053521941, rho: 1.9738402328780484
Iteration 152, prox-grad norm: 0.00249241944567, step size: 208.115127599, rho: 2.03305543986439
Iteration 153, prox-grad norm: 0.00247187045437, step size: 214.358581427, rho: 2.094047103060322
Iteration 154, prox-grad norm: 0.00245137829575, step size: 220.78933887, rho: 2.1568685161521315
Iteration 155, prox-grad norm: 0.00243058467144, step size: 227.413019036, rho: 2.2215745716366957
Iteration 156, prox-grad norm: 0.00240949196613, step size: 234.235409607, rho: 2.2882218087857966
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Achieved relative tolerance at iteration 151

Total running time of the script: ( 0 minutes 3.593 seconds)