Constained MOD¶
This example demonstrates the use of pgm.cmod.CnstrMOD for
computing a dictionary update via a constrained variant of the method of
optimal directions [20]. It also illustrates the
use of pgm.backtrack.BacktrackRobust,
pgm.stepsize.StepSizePolicyCauchy and
pgm.stepsize.StepSizePolicyBB to adapt the step size
parameter of PGM. This problem is mainly useful as a component within
dictionary learning, but its use is demonstrated here since a user may
wish to construct such objects as part of a custom dictionary learning
algorithm, using dictlrn.DictLearn.
from __future__ import print_function
from builtins import input
import numpy as np
from sporco.pgm import bpdn
from sporco.pgm import cmod
from sporco.pgm.backtrack import BacktrackRobust
from sporco.pgm.stepsize import StepSizePolicyBB, StepSizePolicyCauchy
from sporco import util
from sporco import array
from sporco import plot
plot.config_notebook_plotting()
Load training images.
exim = util.ExampleImages(scaled=True, zoom=0.25, gray=True)
S1 = exim.image('barbara.png', idxexp=np.s_[10:522, 100:612])
S2 = exim.image('kodim23.png', idxexp=np.s_[:, 60:572])
S3 = exim.image('monarch.png', idxexp=np.s_[:, 160:672])
S4 = exim.image('sail.png', idxexp=np.s_[:, 210:722])
S5 = exim.image('tulips.png', idxexp=np.s_[:, 30:542])
Extract all 8x8 image blocks, reshape, and subtract block means.
S = array.extract_blocks((S1, S2, S3, S4, S5), (8, 8))
S = np.reshape(S, (np.prod(S.shape[0:2]), S.shape[2]))
S -= np.mean(S, axis=0)
Load initial dictionary.
D0 = util.convdicts()['G:8x8x64']
D0 = np.reshape(D0, (np.prod(D0.shape[0:2]), D0.shape[2]))
Compute sparse representation on current dictionary.
lmbda = 0.1
opt = bpdn.BPDN.Options({'Verbose': True, 'MaxMainIter': 50, 'L': 100,
'Backtrack': BacktrackRobust()})
b = bpdn.BPDN(D0, S, lmbda, opt)
X = b.solve()
Itn Fnc DFid Regℓ1 Rsdl F Q It_Bt L
---------------------------------------------------------------------------------
0 3.96e+04 3.92e+04 3.98e+03 4.20e+00 3.92e+04 4.00e+04 1 9.00e+01
1 3.81e+04 3.73e+04 8.12e+03 8.58e+00 3.73e+04 3.80e+04 1 8.10e+01
2 3.63e+04 3.50e+04 1.35e+04 1.01e+01 3.50e+04 3.56e+04 1 7.29e+01
3 3.44e+04 3.24e+04 1.98e+04 1.13e+01 3.24e+04 3.30e+04 1 6.56e+01
4 3.24e+04 2.97e+04 2.70e+04 1.22e+01 2.97e+04 3.02e+04 1 5.90e+01
5 3.06e+04 2.71e+04 3.47e+04 1.27e+01 2.71e+04 2.75e+04 1 5.31e+01
6 2.90e+04 2.48e+04 4.25e+04 1.29e+01 2.48e+04 2.51e+04 1 4.78e+01
7 2.78e+04 2.28e+04 5.02e+04 1.26e+01 2.28e+04 2.30e+04 1 4.30e+01
8 2.68e+04 2.10e+04 5.74e+04 1.21e+01 2.10e+04 2.12e+04 1 3.87e+01
9 2.60e+04 1.96e+04 6.37e+04 1.15e+01 1.96e+04 1.97e+04 1 3.49e+01
10 2.54e+04 1.84e+04 6.92e+04 1.10e+01 1.84e+04 1.85e+04 1 3.14e+01
11 2.48e+04 1.75e+04 7.37e+04 1.06e+01 1.75e+04 1.75e+04 1 2.82e+01
12 2.44e+04 1.67e+04 7.72e+04 1.04e+01 1.67e+04 1.67e+04 1 2.54e+01
13 2.40e+04 1.60e+04 8.00e+04 1.01e+01 1.60e+04 1.61e+04 1 2.29e+01
14 2.37e+04 1.55e+04 8.21e+04 9.96e+00 1.55e+04 1.56e+04 1 2.06e+01
15 2.35e+04 1.51e+04 8.38e+04 9.77e+00 1.51e+04 1.51e+04 1 1.85e+01
16 2.32e+04 1.47e+04 8.51e+04 9.61e+00 1.47e+04 1.48e+04 1 1.67e+01
17 2.30e+04 1.44e+04 8.62e+04 9.46e+00 1.44e+04 1.45e+04 1 1.50e+01
18 2.29e+04 1.42e+04 8.70e+04 9.33e+00 1.42e+04 1.42e+04 1 1.35e+01
19 2.27e+04 1.40e+04 8.76e+04 9.19e+00 1.40e+04 1.40e+04 1 1.22e+01
20 2.26e+04 1.38e+04 8.81e+04 9.04e+00 1.38e+04 1.38e+04 1 1.09e+01
21 2.25e+04 1.36e+04 8.84e+04 8.88e+00 1.36e+04 1.37e+04 1 9.85e+00
22 2.24e+04 1.35e+04 8.87e+04 8.71e+00 1.35e+04 1.36e+04 1 8.86e+00
23 2.23e+04 1.34e+04 8.88e+04 8.51e+00 1.34e+04 1.35e+04 1 7.98e+00
24 2.23e+04 1.34e+04 8.89e+04 8.29e+00 1.34e+04 1.34e+04 1 7.18e+00
25 2.22e+04 1.33e+04 8.90e+04 8.05e+00 1.33e+04 1.33e+04 1 6.46e+00
26 2.22e+04 1.33e+04 8.90e+04 7.79e+00 1.33e+04 1.33e+04 1 5.81e+00
27 2.21e+04 1.32e+04 8.91e+04 7.50e+00 1.32e+04 1.32e+04 1 5.23e+00
28 2.21e+04 1.32e+04 8.91e+04 7.18e+00 1.32e+04 1.32e+04 1 4.71e+00
29 2.21e+04 1.32e+04 8.91e+04 6.84e+00 1.32e+04 1.32e+04 1 4.24e+00
30 2.21e+04 1.32e+04 8.91e+04 6.47e+00 1.32e+04 1.32e+04 1 3.82e+00
31 2.21e+04 1.32e+04 8.91e+04 6.09e+00 1.32e+04 1.32e+04 1 3.43e+00
32 2.21e+04 1.31e+04 8.91e+04 5.69e+00 1.31e+04 1.31e+04 1 3.09e+00
33 2.20e+04 1.31e+04 8.91e+04 5.28e+00 1.31e+04 1.31e+04 1 2.78e+00
34 2.20e+04 1.31e+04 8.91e+04 4.87e+00 1.31e+04 1.31e+04 1 2.50e+00
35 2.20e+04 1.31e+04 8.91e+04 3.42e+00 1.31e+04 1.31e+04 2 4.51e+00
36 2.20e+04 1.31e+04 8.91e+04 2.89e+00 1.31e+04 1.31e+04 1 4.06e+00
37 2.20e+04 1.31e+04 8.91e+04 2.08e+00 1.31e+04 1.31e+04 2 7.30e+00
38 2.20e+04 1.31e+04 8.91e+04 1.88e+00 1.31e+04 1.31e+04 1 6.57e+00
39 2.20e+04 1.31e+04 8.91e+04 1.81e+00 1.31e+04 1.31e+04 1 5.91e+00
40 2.20e+04 1.31e+04 8.90e+04 1.77e+00 1.31e+04 1.31e+04 1 5.32e+00
41 2.20e+04 1.31e+04 8.90e+04 1.73e+00 1.31e+04 1.31e+04 1 4.79e+00
42 2.20e+04 1.31e+04 8.90e+04 1.68e+00 1.31e+04 1.31e+04 1 4.31e+00
43 2.20e+04 1.31e+04 8.90e+04 1.63e+00 1.31e+04 1.31e+04 1 3.88e+00
44 2.20e+04 1.31e+04 8.90e+04 1.58e+00 1.31e+04 1.31e+04 1 3.49e+00
45 2.20e+04 1.31e+04 8.90e+04 1.52e+00 1.31e+04 1.31e+04 1 3.14e+00
46 2.20e+04 1.31e+04 8.90e+04 1.08e+00 1.31e+04 1.31e+04 2 5.66e+00
47 2.20e+04 1.31e+04 8.90e+04 9.85e-01 1.31e+04 1.31e+04 1 5.09e+00
48 2.20e+04 1.31e+04 8.90e+04 9.69e-01 1.31e+04 1.31e+04 1 4.58e+00
49 2.20e+04 1.31e+04 8.90e+04 9.51e-01 1.31e+04 1.31e+04 1 4.12e+00
---------------------------------------------------------------------------------
Update dictionary for training image set using PGM with Cauchy step size policy [63].
opt = cmod.CnstrMOD.Options({'Verbose': True, 'MaxMainIter': 100, 'L': 50,
'StepSizePolicy': StepSizePolicyCauchy()})
c1 = cmod.CnstrMOD(X, S, None, opt)
D11 = c1.solve()
print("CMOD solve time: %.2fs" % c1.timer.elapsed('solve'))
Itn DFid Cnstr Rsdl
----------------------------------
0 1.35e+04 7.54e-07 7.94e+00
1 4.85e+04 6.92e-07 8.91e+00
2 1.87e+04 7.50e-07 5.84e+00
3 1.42e+04 7.32e-07 1.98e+00
4 1.25e+04 7.19e-07 9.50e-01
5 1.19e+04 8.14e-07 6.70e-01
6 1.15e+04 6.55e-07 5.04e-01
7 1.13e+04 7.86e-07 4.55e-01
8 1.13e+04 7.51e-07 4.08e-01
9 1.12e+04 7.25e-07 3.87e-01
10 1.12e+04 6.79e-07 3.74e-01
11 1.12e+04 7.24e-07 3.68e-01
12 1.12e+04 7.31e-07 3.64e-01
13 1.12e+04 7.64e-07 3.61e-01
14 1.12e+04 7.86e-07 3.60e-01
15 1.12e+04 7.58e-07 3.59e-01
16 1.12e+04 8.34e-07 3.59e-01
17 1.12e+04 7.82e-07 3.60e-01
18 1.12e+04 7.43e-07 3.62e-01
19 1.12e+04 8.41e-07 3.64e-01
20 1.12e+04 8.25e-07 3.66e-01
21 1.12e+04 7.26e-07 3.68e-01
22 1.12e+04 7.84e-07 3.70e-01
23 1.12e+04 7.42e-07 3.71e-01
24 1.12e+04 6.80e-07 3.73e-01
25 1.12e+04 7.14e-07 3.74e-01
26 1.12e+04 7.89e-07 3.75e-01
27 1.12e+04 7.09e-07 3.76e-01
28 1.12e+04 6.14e-07 3.77e-01
29 1.12e+04 7.83e-07 3.77e-01
30 1.12e+04 6.96e-07 3.78e-01
31 1.12e+04 8.06e-07 3.79e-01
32 1.12e+04 7.46e-07 3.80e-01
33 1.12e+04 7.61e-07 3.80e-01
34 1.12e+04 7.01e-07 3.81e-01
35 1.12e+04 7.50e-07 3.82e-01
36 1.12e+04 6.73e-07 3.82e-01
37 1.12e+04 8.36e-07 3.83e-01
38 1.12e+04 7.36e-07 3.84e-01
39 1.12e+04 7.32e-07 3.84e-01
40 1.12e+04 7.45e-07 3.85e-01
41 1.12e+04 6.96e-07 3.85e-01
42 1.12e+04 6.66e-07 3.85e-01
43 1.12e+04 7.17e-07 3.86e-01
44 1.12e+04 7.77e-07 3.86e-01
45 1.12e+04 6.81e-07 3.86e-01
46 1.12e+04 6.82e-07 3.87e-01
47 1.12e+04 8.05e-07 3.87e-01
48 1.12e+04 7.03e-07 3.87e-01
49 1.12e+04 8.16e-07 3.88e-01
50 1.12e+04 7.70e-07 3.88e-01
51 1.12e+04 8.00e-07 3.88e-01
52 1.12e+04 7.73e-07 3.88e-01
53 1.12e+04 7.86e-07 3.89e-01
54 1.12e+04 8.63e-07 3.89e-01
55 1.12e+04 8.40e-07 3.89e-01
56 1.12e+04 6.29e-07 3.89e-01
57 1.12e+04 7.65e-07 3.90e-01
58 1.12e+04 8.17e-07 3.90e-01
59 1.12e+04 7.49e-07 3.90e-01
60 1.12e+04 8.23e-07 3.90e-01
61 1.12e+04 8.16e-07 3.90e-01
62 1.12e+04 7.66e-07 3.91e-01
63 1.12e+04 7.67e-07 3.91e-01
64 1.12e+04 7.02e-07 3.91e-01
65 1.12e+04 7.69e-07 3.91e-01
66 1.12e+04 7.68e-07 3.91e-01
67 1.12e+04 7.80e-07 3.91e-01
68 1.12e+04 7.85e-07 3.92e-01
69 1.12e+04 8.74e-07 3.92e-01
70 1.12e+04 7.83e-07 3.92e-01
71 1.12e+04 8.02e-07 3.92e-01
72 1.12e+04 7.88e-07 3.92e-01
73 1.12e+04 6.95e-07 3.92e-01
74 1.12e+04 6.73e-07 3.92e-01
75 1.12e+04 7.36e-07 3.93e-01
76 1.12e+04 7.83e-07 3.93e-01
77 1.12e+04 7.15e-07 3.93e-01
78 1.12e+04 7.82e-07 3.93e-01
79 1.12e+04 7.80e-07 3.93e-01
80 1.12e+04 8.70e-07 3.93e-01
81 1.12e+04 6.41e-07 3.93e-01
82 1.12e+04 6.15e-07 3.93e-01
83 1.12e+04 6.91e-07 3.93e-01
84 1.12e+04 8.60e-07 3.94e-01
85 1.12e+04 7.66e-07 3.94e-01
86 1.12e+04 7.53e-07 3.94e-01
87 1.12e+04 8.72e-07 3.94e-01
88 1.12e+04 8.48e-07 3.94e-01
89 1.12e+04 7.22e-07 3.94e-01
90 1.12e+04 7.31e-07 3.94e-01
91 1.12e+04 7.81e-07 3.94e-01
92 1.12e+04 7.60e-07 3.94e-01
93 1.12e+04 6.93e-07 3.94e-01
94 1.12e+04 8.34e-07 3.94e-01
95 1.12e+04 7.30e-07 3.95e-01
96 1.12e+04 7.29e-07 3.95e-01
97 1.12e+04 8.26e-07 3.95e-01
98 1.12e+04 8.32e-07 3.95e-01
99 1.12e+04 7.19e-07 3.95e-01
----------------------------------
CMOD solve time: 4.75s
Update dictionary for training image set using PGM with Barzilai-Borwein step size policy [4].
opt = cmod.CnstrMOD.Options({'Verbose': True, 'MaxMainIter': 100, 'L': 50,
'StepSizePolicy': StepSizePolicyBB()})
c2 = cmod.CnstrMOD(X, S, None, opt)
D12 = c2.solve()
print("CMOD solve time: %.2fs" % c2.timer.elapsed('solve'))
Itn DFid Cnstr Rsdl
----------------------------------
0 1.35e+04 7.54e-07 7.94e+00
1 4.85e+04 6.92e-07 8.91e+00
2 1.95e+04 6.95e-07 5.65e+00
3 1.56e+04 7.75e-07 1.33e+00
4 1.41e+04 6.97e-07 3.28e-01
5 1.23e+04 7.38e-07 1.19e+00
6 1.16e+04 8.11e-07 5.47e-01
7 1.14e+04 7.59e-07 2.18e-01
8 1.13e+04 8.10e-07 2.70e-01
9 1.12e+04 8.00e-07 1.75e-01
10 1.12e+04 8.11e-07 1.15e-01
11 1.12e+04 7.75e-07 6.27e-02
12 1.12e+04 7.97e-07 5.79e-02
13 1.12e+04 6.95e-07 2.37e-01
14 1.12e+04 7.57e-07 9.25e-02
15 1.12e+04 8.52e-07 5.03e-02
16 1.12e+04 7.76e-07 1.78e-02
17 1.12e+04 7.46e-07 5.30e-02
18 1.12e+04 7.37e-07 5.39e-02
19 1.12e+04 8.67e-07 4.21e-02
20 1.12e+04 7.26e-07 3.32e-02
21 1.12e+04 7.71e-07 2.03e-02
22 1.12e+04 7.01e-07 8.57e-03
23 1.12e+04 6.74e-07 7.92e-02
24 1.12e+04 8.06e-07 2.38e-02
25 1.12e+04 7.77e-07 6.17e-03
26 1.12e+04 7.57e-07 1.04e-01
27 1.12e+04 6.80e-07 5.13e-02
28 1.12e+04 7.87e-07 6.63e-03
29 1.12e+04 8.10e-07 2.80e-02
30 1.12e+04 6.90e-07 2.08e-02
31 1.12e+04 6.55e-07 3.90e-03
32 1.12e+04 8.58e-07 6.01e-02
33 1.12e+04 7.59e-07 3.11e-02
34 1.12e+04 7.11e-07 8.03e-03
35 1.12e+04 7.95e-07 5.28e-02
36 1.12e+04 6.53e-07 1.71e-02
37 1.12e+04 7.48e-07 3.14e-03
38 1.12e+04 5.70e-07 1.57e-02
39 1.12e+04 7.51e-07 8.81e-03
40 1.12e+04 6.48e-07 2.97e-03
41 1.12e+04 6.86e-07 2.18e-02
42 1.12e+04 7.35e-07 1.15e-02
43 1.12e+04 8.12e-07 1.46e-03
44 1.12e+04 7.96e-07 3.06e-02
45 1.12e+04 8.51e-07 2.12e-02
46 1.12e+04 7.79e-07 1.18e-02
47 1.12e+04 8.06e-07 3.44e-03
48 1.12e+04 8.40e-07 7.64e-03
49 1.12e+04 8.52e-07 1.65e-03
50 1.12e+04 7.65e-07 1.34e-02
51 1.12e+04 7.14e-07 7.18e-03
52 1.12e+04 6.94e-07 2.73e-04
----------------------------------
CMOD solve time: 1.64s
Display initial and final dictionaries.
D0 = D0.reshape((8, 8, D0.shape[-1]))
D11 = D11.reshape((8, 8, D11.shape[-1]))
D12 = D12.reshape((8, 8, D12.shape[-1]))
fig = plot.figure(figsize=(14, 7))
plot.subplot(1, 3, 1)
plot.imview(util.tiledict(D0), title='D0', fig=fig)
plot.subplot(1, 3, 2)
plot.imview(util.tiledict(D11), title='D1 Cauchy', fig=fig)
plot.subplot(1, 3, 3)
plot.imview(util.tiledict(D12), title='D1 BB', fig=fig)
fig.show()
Get iterations statistics from CMOD solver object and plot functional value, residuals, and automatically adjusted L against the iteration number.
its1 = c1.getitstat()
its2 = c2.getitstat()
fig = plot.figure(figsize=(20, 5))
plot.subplot(1, 3, 1)
plot.plot(its1.DFid, xlbl='Iterations', ylbl='Functional', fig=fig)
plot.plot(its2.DFid, xlbl='Iterations', ylbl='Functional',
lgnd=['Cauchy', 'BB'], fig=fig)
plot.subplot(1, 3, 2)
plot.plot(its1.Rsdl, ptyp='semilogy', xlbl='Iterations', ylbl='Residual',
fig=fig)
plot.plot(its2.Rsdl, ptyp='semilogy', xlbl='Iterations', ylbl='Residual',
lgnd=['Cauchy', 'BB'], fig=fig)
plot.subplot(1, 3, 3)
plot.plot(its1.L, xlbl='Iterations', ylbl='Inverse of Step Size', fig=fig)
plot.plot(its2.L, xlbl='Iterations', ylbl='Inverse of Step Size',
lgnd=[r'$L_{Cauchy}$', '$L_{BB}$'], fig=fig)
fig.show()
