Signal Processing - November 2017 - 91
number of parameters [23]; constraining the filters to be sepaand regularizer, g. For a textbook coverage of the subject of
rable [12] further reduces the number of parameters. Doing this
learning, see [33].
can give the experimenter a sense of the training time required
Understanding the learning minimization problem as a
on their hardware as well as the effects of the network size on
statistical inference can provide useful insight into the selecperformance. From this simple starting point, the architecture
tion of the cost and regularization functions. From this percan be tweaked for greater performance; for example, by addspective, we can formulate the goal of learning as maximizing
ing downsampling and upsampling operations [27] or by simply
the conditional likelihood of each training output given the
adding more layers [20].
corresponding training input and CNN parameters:
Instead of using ad hoc architecture design, one can adapt
given {(x n, y n)} nN= 1,
a successful CNN architecture from another application. For
N
example, [27] adapts a network designed for biomedical image
R learn = arg max % P (y n | x n, i),
segmentation to CT reconstruction by changing the number of
R ,i ! H n =1
output layers from two (background and foreground images) to
one (reconstructed image). These architectures can also be conwhere P is a conditional likelihood. When this likelihood folnected end to end, creating modular or hierarchical designs. For
lows a Gaussian distribution, this optimization is equivalent
example, a four-times superresolution architecture can be created
to the one from the "Background" section, (3), with f being
by connecting two two-times superresolution
the Euclidean distance and no regularizanetworks [16]. This is distinct from training
tion. Put another way, learning with the
The notion of universal
a two-times superresolution network and
standard, Euclidean cost, and no regularapproximation tells us
applying it twice because the two modules of
ization implicitly assumes a Gaussian noise
what the network can
the CNN are trained as a unit.
model; this is a well-known fact in inverse
learn, not what it does
A second approach is to begin with an
problems in general. This formulation
iterative optimization algorithm and unroll it,
is used in most of the works we surveyed
learn, and comparison
turning each iteration into a layer of a net[6], [7], [11], [12], [18], [19], [23], [25], [26],
to established algorithms
work. In such a scheme, filters that are nordespite the fact that several raise questions
can help guide our
mally fixed in the iterative minimization are
about whether it is the best choice [25], [34].
understanding of CNNs
instead learned. The approach was pioneered
An alternative is the maximum a posin practice.
in [31] for sparse coding; their results showed
teriori formulation, which maximizes the
that the learned algorithms could achieve a
joint probability of the training data and
given error in fewer iterations than the standard ones. Because
the CNN parameters, which can be decomposed into several
many iterative optimization algorithms alternate filtering
terms using Bayes' rule:
steps (linear updates) with pointwise nonlinear steps (proximal/
given {(x n, y n)} nN= 1,
shrinkage operations), the resulting network is often a CNN. This
N
was the approach in [22], where the authors unrolled the alternat
R learn = arg max % P (y n | x n, i) P (i) .
(4)
ing direction method of multipliers (ADMM) algorithm to design
R ,i ! H n =1
a CNN for MRI reconstruction, with state-of-the-art results and
improvements in running time. For networks designed in this
This formulation explicitly allows prior information about the
way, the original algorithm is a specific case, and, therefore, the
desired CNN parameters, i, to be used. Under a Gaussian
performance of the network cannot be worse than the original
model for the weights of the CNN as well as the noise, this foralgorithm if training is successful. The concept of unrolling can
mulation results in a Euclidean cost function and a Euclidean
also be applied at a coarser scale, as in [13], where the modules
regularization on the weights of the CNN, g (i) = v -2 | | i | | 22 .
of the network mimic the steps of a typical blind deconvolution
Other examples of regularizations for CNNs are the total genpipeline: extract features, estimate kernel, estimate image, repeat.
eralized variation norm or sparsity on the coefficients. ReguAnother promising design approach, similar to unrolling,
larized approaches are taken in [10], [15], [21], and [22].
is to learn only some part of an existing iterative method. For
example, given the modular nature of popular iterative optimiOptimization
zation schemes such as the ADMM, a CNN can be employed
Once an objective function for learning has been fixed, it still
as a proximal (denoising) operator, while the rest of the algomust be actually minimized. This is a crucial and deep topic,
rithm remains unchanged [32]. This design combines many of
but, from the practical perspective, it can be treated as a black
the good aspects of both the objective function and learningbox due to the availability of several high-quality software
based approaches and allows a single CNN to be used for sevlibraries that can perform efficient training of user-defined
eral different inverse problems without retraining.
CNN architectures. For a comparison of these libraries, refer
to [35]; here, we provide a basic overview.
The popular approaches to CNN learning are variations
Cost function and regularization
on gradient descent. The most common is stochastic gradiIn this section, we survey the approaches taken to actually
ent descent (SGD), used, e.g., in [16] and [25], where, at each
train the CNN, including the choice of a cost function, f,
i
i
IEEE SIGNAL PROCESSING MAGAZINE
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November 2017
|
91
Table of Contents for the Digital Edition of Signal Processing - November 2017
Signal Processing - November 2017 - Cover1
Signal Processing - November 2017 - Cover2
Signal Processing - November 2017 - 1
Signal Processing - November 2017 - 2
Signal Processing - November 2017 - 3
Signal Processing - November 2017 - 4
Signal Processing - November 2017 - 5
Signal Processing - November 2017 - 6
Signal Processing - November 2017 - 7
Signal Processing - November 2017 - 8
Signal Processing - November 2017 - 9
Signal Processing - November 2017 - 10
Signal Processing - November 2017 - 11
Signal Processing - November 2017 - 12
Signal Processing - November 2017 - 13
Signal Processing - November 2017 - 14
Signal Processing - November 2017 - 15
Signal Processing - November 2017 - 16
Signal Processing - November 2017 - 17
Signal Processing - November 2017 - 18
Signal Processing - November 2017 - 19
Signal Processing - November 2017 - 20
Signal Processing - November 2017 - 21
Signal Processing - November 2017 - 22
Signal Processing - November 2017 - 23
Signal Processing - November 2017 - 24
Signal Processing - November 2017 - 25
Signal Processing - November 2017 - 26
Signal Processing - November 2017 - 27
Signal Processing - November 2017 - 28
Signal Processing - November 2017 - 29
Signal Processing - November 2017 - 30
Signal Processing - November 2017 - 31
Signal Processing - November 2017 - 32
Signal Processing - November 2017 - 33
Signal Processing - November 2017 - 34
Signal Processing - November 2017 - 35
Signal Processing - November 2017 - 36
Signal Processing - November 2017 - 37
Signal Processing - November 2017 - 38
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Signal Processing - November 2017 - 40
Signal Processing - November 2017 - 41
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Signal Processing - November 2017 - 43
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Signal Processing - November 2017 - 47
Signal Processing - November 2017 - 48
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Signal Processing - November 2017 - 60
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Signal Processing - November 2017 - 68
Signal Processing - November 2017 - 69
Signal Processing - November 2017 - 70
Signal Processing - November 2017 - 71
Signal Processing - November 2017 - 72
Signal Processing - November 2017 - 73
Signal Processing - November 2017 - 74
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Signal Processing - November 2017 - 76
Signal Processing - November 2017 - 77
Signal Processing - November 2017 - 78
Signal Processing - November 2017 - 79
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Signal Processing - November 2017 - 81
Signal Processing - November 2017 - 82
Signal Processing - November 2017 - 83
Signal Processing - November 2017 - 84
Signal Processing - November 2017 - 85
Signal Processing - November 2017 - 86
Signal Processing - November 2017 - 87
Signal Processing - November 2017 - 88
Signal Processing - November 2017 - 89
Signal Processing - November 2017 - 90
Signal Processing - November 2017 - 91
Signal Processing - November 2017 - 92
Signal Processing - November 2017 - 93
Signal Processing - November 2017 - 94
Signal Processing - November 2017 - 95
Signal Processing - November 2017 - 96
Signal Processing - November 2017 - 97
Signal Processing - November 2017 - 98
Signal Processing - November 2017 - 99
Signal Processing - November 2017 - 100
Signal Processing - November 2017 - 101
Signal Processing - November 2017 - 102
Signal Processing - November 2017 - 103
Signal Processing - November 2017 - 104
Signal Processing - November 2017 - 105
Signal Processing - November 2017 - 106
Signal Processing - November 2017 - 107
Signal Processing - November 2017 - 108
Signal Processing - November 2017 - 109
Signal Processing - November 2017 - 110
Signal Processing - November 2017 - 111
Signal Processing - November 2017 - 112
Signal Processing - November 2017 - 113
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Signal Processing - November 2017 - 115
Signal Processing - November 2017 - 116
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Signal Processing - November 2017 - 118
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Signal Processing - November 2017 - 120
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Signal Processing - November 2017 - 125
Signal Processing - November 2017 - 126
Signal Processing - November 2017 - 127
Signal Processing - November 2017 - 128
Signal Processing - November 2017 - 129
Signal Processing - November 2017 - 130
Signal Processing - November 2017 - 131
Signal Processing - November 2017 - 132
Signal Processing - November 2017 - 133
Signal Processing - November 2017 - 134
Signal Processing - November 2017 - 135
Signal Processing - November 2017 - 136
Signal Processing - November 2017 - 137
Signal Processing - November 2017 - 138
Signal Processing - November 2017 - 139
Signal Processing - November 2017 - 140
Signal Processing - November 2017 - 141
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Signal Processing - November 2017 - 144
Signal Processing - November 2017 - 145
Signal Processing - November 2017 - 146
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Signal Processing - November 2017 - 148
Signal Processing - November 2017 - 149
Signal Processing - November 2017 - 150
Signal Processing - November 2017 - 151
Signal Processing - November 2017 - 152
Signal Processing - November 2017 - 153
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Signal Processing - November 2017 - 155
Signal Processing - November 2017 - 156
Signal Processing - November 2017 - 157
Signal Processing - November 2017 - 158
Signal Processing - November 2017 - 159
Signal Processing - November 2017 - 160
Signal Processing - November 2017 - 161
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Signal Processing - November 2017 - 167
Signal Processing - November 2017 - 168
Signal Processing - November 2017 - 169
Signal Processing - November 2017 - 170
Signal Processing - November 2017 - 171
Signal Processing - November 2017 - 172
Signal Processing - November 2017 - 173
Signal Processing - November 2017 - 174
Signal Processing - November 2017 - 175
Signal Processing - November 2017 - 176
Signal Processing - November 2017 - Cover3
Signal Processing - November 2017 - Cover4
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