Digital Signal Processing Reference
In-Depth Information
The following result relates the proximity operator of
F
and that of its conjugate
F
∗
.Let
F
∈
0
(
H
); then, for any
α
∈H
,
prox
F
∗
(
α
)
+
ρ
prox
F
/ρ
(
α/ρ
)
=
α,
∀
0
<ρ<
+∞
.
(7.11)
ρ
The first proof of this result dates back to Moreau (1965) for
1, and other ones
are in Rockafellar (1970, Theorem 31.5) and Combettes and Wajs (2005, Lemma
2.10). From equation (7.11), we conclude that
ρ
=
prox
F
∗
=
I
−
prox
F
.
7.3.2.2 Proximity Operator of Convex Sparsity Penalties
In the following, we consider a family of sparsity-promoting potential functions, in
which the proximity operator takes an appealing closed form. Toward this goal,
consider the additive sparsity promoting penalty:
T
(
α
)
=
1
ψ
i
(
α
[
i
])
,
i
=
where we assume that
∀
1
≤
i
≤
T
.
Assumption 1
1.
ψ
i
∈
0
(
R
)
.
2.
ψ
i
is convex, even-symmetric, nonnegative, and nondecreasing on
[0
,
+∞
)
.
3.
ψ
i
is continuous on
R
, with
ψ
i
(0)
=
0
.
4.
ψ
i
differentiable on
(0
)
but is not necessarily smooth at
0
and admits a
positive right derivative at zero
,
+∞
i
lim
t
↓
0
ψ
(
t
)
ψ
(0)
=
t
≥
0
.
+
0, has exactly
one continuous and odd-symmetric solution decoupled in each coordinate
i
:
Under Assumption 1(1)-(4), the proximity operator of
κ
ψ
i
,
κ>
0
if
α
[
i
]
≤
κ
ψ
i
(0)
if
α
[
i
]
>κ
ψ
+
α
˜
[
i
]
=
prox
κ
ψ
i
(
α
[
i
])
=
(0)
.
(7.12)
i
(˜
i
α
[
i
]
−
κ
ψ
α
[
i
])
+
The proof can be found in Fadili and Starck (2009) and uses the calculus rule of
Section 7.3.2.1(4). Similar results to equation (7.12) appear under different forms
in several papers, including Antoniadis and Fan (2001), Nikolova (2000), Fadili and
Bullmore (2005), Fadili et al. (2009c), and Combettes and Pesquet (2007b). Some of
these works dealt even with nonconvex penalties (see Section 7.7.3 and the discus-
sion at the end of Section 6.2.2). This result covers the popular case
(
α
)
=
λ
α
1
,
where the solution is given by soft thresholding:
1
[
i
]
−
λ
α
[
i
]
prox
λ
·
1
(
α
)
=
SoftThresh
(
α
)
=
α
T
,
(7.13)
λ
+
1
≤
i
≤
where (
·
)
+
=
max(
·
,
0). See equation (6.14).
