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5
LTS2 - EPFL
5
PROXIMAL OPERATORS
Proximal operators
5.1
Presentation of different norms
Usual proximal operator minimize norm. In the following table, we present the main of them.
For a continuous signal x(t) defined on a open set Ω ⊂ Rp we define different norms in table 2.
For a discrete signal x(n) with n ∈ 1, 2, ..., N , we also define norms in table 2.
Name
Notation
Matlab Function
L1 norm
`1
norm(·,1)
Continuous definition
Discrete definition
Z
kxk1 =
|x(t)|dt
t∈Ω
kxk1 =
N
X
|xn |
n=1
L2 norm
`2
norm(·)
s
Z
v
uN
uX
kxk2 = t
|xn |2
|x(t)|2 dt
kxk2 =
t∈Ω
n=1
L12 mixed norm
`12
norm_L12(·,g_d,g_t)
Z
Z
2
21
|x(t1 , t2 )| dt2
kxk12 =
t1 ∈Ω1
dt1
kxk12 =
X
kxg k2 ,
g∈G
t2 ∈Ω2
with xg ⊂ x
L1∞ mixed norm
`1∞
norm_L1inf(·,g_d,g_t)
Z
kxk1∞ =
max |x(t1 , t2 )| dt1
kxk1∞ =
t2 ∈Ω2
t1 ∈Ω1
X
kxg k∞ ,
g∈G
with xg ⊂ x
TV norm
`T V
TV_norm(·)
TV_norm3D(·)
Z
or
kxkT V =
|∇x(t)|dt
t∈Ω
kxkT V =
X
|∇x(n)|
n
For x a m byn matrix
with singular values σi ,
we have:
Nuclear norm
`∗
norm_nuclear(·)
min{m,n}
kxk∗ =
X
σi
i=1
Table 2: Presentation of the norms
5.2
Norm proximal operators
Minimizing norm presented in table 2 leads to common proximal operators. We present them
in this section. All Matlab prox functions take three parameters: x, lambda, param. First, x is
the initial signal. Then lambda is the weight of the objective function. Finally, param is a Matlab
structure that containing a set of optional parameters.
July 2012
6/16