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min
Z
+
ʻ
E
,
2,1
(1)
Z
*
st X
..,
=+
AZ
E
,
[
]
Z
=
zz
1
,,,
n
is the coefficient matrix with each
z
where
z
being the represen-
[
]
tation of
x
,
Aaa
=
1
,,,
m
is the overcomplete dictionary which can represent
a
[
( )
2
is called as
n
n
x
by the linear combination of its basis,
E
=
E
2,1
j
=
1
i
=
1
ij
2,
-norm, and the parameter
the
ʻ >
0
is used to balance the effects of the two
⇅
parts. Here,
denotes the
nuclear norm
[12] of a matrix,
i.e.
, the sum of the singu-
lar values of the matrix.
Using the data
X
itself as the dictionary, Eq. (1) can be converted into:
*
min
Z
+
ʻ
E
,
2,1
(2)
Z
*
st X
..,
=+
XZ
E
,
where
XZ
is the low-rank matrix, and
E
is the sparse matrix.
2.2
Solution to the Optimization Problem
For Eq. (2), it can be viewed as the following equivalent problem:
min
ZEJ
J
∗
+
=+
=
ʻ
E
,
2,1
,,
st X
.,
XZ
E
,
(3)
ZJ
,
which can be solved by Augmented Lagrange Multiplier (ALM) method [13]:
min
J
∗
+
ʻ
E
+
2,1
ZEJ
,,
(
)
(
)
t
t
tr
YXXZE
−−+
tr
YZJ
−+
(4)
1
2
μ
(
)
2
2
XXZE
−− +−
ZJ
,
2
F
F
μ >
is a penalty parameter. We
outline the inexact ALM in Algorithm 1. Note that Steps 1 and 3 of the algorithm are
convex problems. However, both of them have closed-form solutions. Particularly
Step 1 is solved via singular value thresholding operator [14], and Step 3 can be
solved by the following lemma [9]:
Lemma
Let
0
where
Y
and
Y
are Lagrange multipliers and
[
]
Qqq
=
1
,,,,
be a given matrix and
q
⇅
be the Frobenius
i
F
norm. If the optimal solution of
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