Digital Signal Processing Reference
In-Depth Information
Section 7.3.2.1(4), it can be shown that
1
−
λ
b
α
I
b
prox
b
=
1
λ
b
(
α
)
=
α
b
B
.
(7.15)
·
I
b
+
1
≤
b
≤
7.3.2.3 Precomposition with an Affine Operator
In this section, we provide an important result on the proximity operator of the
precomposition of
F
∈
0
(
H
) with a bounded affine operator
A
:
H
1
→H
2
,α
→
F
y
. It will be at the heart of many algorithms in the rest of the chapter. First,
if
F
is an orthobasis, it can be easily shown from Definition 1 and Section 7.3.2.1(1)
that
α
−
F
∗
prox
F
(
F
prox
F
◦
A
(
α
)
=
y
+
α
−
y
)
.
(7.16)
Otherwise, let
F
be a bounded linear operator. We distinguish three situations:
1.
F
is a tight frame with constant
c
. Then,
F
◦
A
∈
0
(
H
1
) and
c
−
1
F
∗
(prox
cF
−
prox
F
◦
A
(
α
)
=
α
+
I
) (
F
α
−
y
)
.
(7.17)
2.
F
is a general frame with lower and upper bounds
c
1
and
c
2
. Thus
F
◦
A
∈
0
(
H
1
). Define the scheme described in Algorithm 20.
Algorithm 20
Forward-Backward Scheme to Compute the Proximity Operator of
Precomposition with an Affine Operator
Initialization:
Choose some
u
(0)
dom(
F
∗
) and set
p
(0)
F
∗
u
(0)
,
∈
=
α
−
μ
t
∈
(0
c
2
).
Main iteration:
for
t
,
2
/
=
0
to
N
iter
−
1
do
=
μ
t
I
μ
−
t
F
A
p
(
t
)
u
(
t
+
1)
μ
−
t
u
(
t
)
−
prox
+
,
(7.18)
p
(
t
+
1)
F
∗
u
(
t
+
1)
=
α
−
.
:
p
(
N
iter
)
.
Output:
Proximity operator of
F
◦
A
at
α
Then
p
(
t
)
converges linearly to prox
F
◦
A
(
α
), and the best convergence rate
is attained for
μ
t
≡
2
/
(
c
1
+
c
2
).
c
2
2
t
)
)
−
c
1
2
c
2
c
1
2
p
(
t
)
p
(0)
−
α
≤
−
α
prox
F
◦
A
(
prox
F
◦
A
(
(7.19)
+
c
2
c
1
◦
∈
H
3. For all other cases, suppose that
F
A
0
(
1
) and apply Algorithm 20 with
c
2
). Then,
p
(
t
)
converges to prox
F
◦
A
(
μ
t
∈
(0
,
2
/
α
) at the rate
O
(1
/
t
), that is,
∃
C
>
0 such that
)
2
p
(
t
)
−
prox
F
◦
A
(
α
≤
C
/
t
.
(7.20)
