F. Ye et al.
DOI:
10.4236/apm.2018.82008 157 Advances in Pure Mathematics
And we write it in the form of a matrix expression:
( )
YX
θβ ε
= +
(2)
where
[ ]
T
1
,,
N
Yyy=
represents the observed value.
[ ]
T
1
,,
k
θθθ
=
is un-
known parameters represent the frequency.
( ) ( )
Nk
X Nk
θ
∗
determined by
parameter
θ
. The covariate in the model
[ ]
T
1
,,
k
βββ
=
represents the am-
plitude of the corresponding frequency.
[ ]
T
1
,,
n
εεε
=
represents a random
error term, assuming that they are independent.
2.2. Penalty Least Squares Regression
In the process of signal reconstruction, the dimension of
Y
is much smaller than
the number of measurements (
Nk
). Since the signal is sparse, the Equation
(2) would be transformed into an optimization problem (3):
( )
0
min
s.t YX
β
θβ ε
= +
(3)
where
0
β
stands for the number of the non-zero components of
β
. The
optimization (3), however, is an NP-hard problem (which is difficult to find the
solution in polynomial time). We can transform optimization (3) into a penalty
least squares problem:
( )
( )
min
s.t
G
YX
β
θβ ε
= +
(4)
The optimization (4) can be formulated as an unconstrained optimization
problem by removing the constraint and adding a penalty term to the objective
function:
( ) ( )
( )
2
2
,
1
min ,
k
i
i
H YX G
βθ
θβ θ β λ β
=
=−+
∑
(5)
where
λ
represents the adjustable penalty parameter. Different penalty func-
tions form different regularization parameters in the iterative process. We find
that the penalty function of the inverse trigonometric function has better prop-
erties than other common penalty function such as logarithmic penalty function.
In the next section we propose an iterative reweighted
1
l
sparse algorithm with
anti-trigonometric function penalties.
3. Iterative Reweighted l
1
Sparse Algorithm
3.1. Algorithm Description
We now develop an iterative reweighted algorithm for joint dictionary parame-
ter learning and sparse signal recovery. Consider the line spectral estimation
with anti-trigonometric penalty function:
( ) ( )
( )
2
2
,
1
min , arctan
k
i
i
H YX
βθ
θβ θ β λ ϕβ
=
=−+
∑
(6)
We consider the first derivative of the problem (6). Since the absolute value is
involved, we summarize the following derivative functions:
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