IEEE Signal Processing - July 2018 - 84
appearing again in the CSC. The pursuit task is essentially a projection operation, seeking the signal closest to the given data
while belonging to the model, be it Sparseland, the CSC model,
or the ML-CSC. When dealing with a signal X believed to
belong to the ML-CSC model, a noisy signal Y = X + E
^ E 2 # e h is projected to the model by solving the following
pursuit problem:
Find
" C i ,Ki = 1
Z
] Y - D1 C1 2 # e
]
] C1 = D2 C2
s.t. [ C 2 = D 3 C 3
]
h
]
] CK - 1 = DK CK
\
,
,
,
,
_
# k1 b
b
# k2 b
# k3 ` .
b
b
s
C K 0, 3 # k K b
a
(18)
s
0, 3
s
C 2 0, 3
s
C 3 0, 3
C1
We refer to this as the deep coding problem (DCP). Figure 7
illustrates a typical result of such a problem, still in the context
of the MNIST database. As before, we present these without
diving into the details of how this solution was obtained. See
[75] for this information. As can be seen, the very same signal
X is created by sparsely combining 209 atoms from D 1, or
very sparsely combining 47 molecules from D 1 D 2, or
extremely sparsely merging ten body parts from the dictionary
D 1 D 2 D 3.
The DCP problem is NP-hard. Indeed, a single layer version
of it is known to be NP-hard [76], and the additional constraints
can be merged into an overall problem that can be given the same
y
structure as the single layer one, thus exposing its NP-hardness.
Therefore, just as in the general atom-decomposition problem
discussed in the section "Sparse Modeling," approximation algorithms are required for its solution. We are interested in making
our first steps in developing such a pursuit algorithm, and thus our
starting point is an adaptation of the thresholding algorithm, due
to its simplicity.
For a single-layer CSC model (Y = D 1 C 1 + E where
s
C 1 0, 3 # k 1), the thresholding algorithm multiplies Y by D T1 ,
and applies a simple shrinkage on the resulting vector, nulling small entries and leaving the rest almost untouched. This
shrinkage can admit one of several forms, as shown in Figure 8: hard thresholding, soft thresholding, and one-sided soft
thresholding, if the representation C 1 is assumed to be nonnegative. Put formally, this implies that the estimated representation is given by
Ct 1 = TT1 ^D T1 Yh,
where the operator TT1 (V) operates element-wise, thresholding
the entries of the vector V by the values found in the vector T1.
As we add more layers to the CSC model, we can apply the
thresholding algorithm sequentially, each layer at a time. Thus,
once C1 has been estimated, we can now apply a second thresholding algorithm to evaluate C2 by
t 2 = TT2 ^D T2 C
t 1h
C
= TT2 ^D T2 TT1 ^D T1 Yhh .
∧
x
D1
D2
∧
Γ1
D3
∧
Γ2
∧
Γ3
Figure 7. Decompositions of an image from MNIST in terms of its nested sparse features c i and multilayer convolutional dictionaries D i .
84
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IEEE Signal Processing Magazine
|
July 2018
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Table of Contents for the Digital Edition of IEEE Signal Processing - July 2018
Contents
IEEE Signal Processing - July 2018 - Cover1
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IEEE Signal Processing - July 2018 - Cover3
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