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The Journal of fourier analysis and applications, 2020-08, Vol.26 (4), Article 59
Ort / Verlag
New York: Springer US
Erscheinungsjahr
2020
Link zum Volltext
Quelle
SpringerLink
Beschreibungen/Notizen
The aim of this paper is twofold. First, we show that a certain concatenation of a proximity operator with an affine operator is again a proximity operator on a suitable Hilbert space. Second, we use our findings to establish so-called proximal neural networks (PNNs) and stable tight frame proximal neural networks. Let
H
and
K
be real Hilbert spaces,
b
∈
K
and
T
∈
B
(
H
,
K
)
a linear operator with closed range and Moore–Penrose inverse
T
†
. Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator
Prox
:
K
→
K
the operator
T
†
Prox
(
T
·
+
b
)
is a proximity operator on
H
equipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator
Prox
=
S
λ
:
ℓ
2
→
ℓ
2
and any frame analysis operator
T
:
H
→
ℓ
2
that the frame shrinkage operator
T
†
S
λ
T
is a proximity operator on a suitable Hilbert space. The concatenation of proximity operators on
R
d
equipped with different norms establishes a PNN. If the network arises from tight frame analysis or synthesis operators, then it forms an averaged operator. In particular, it has Lipschitz constant 1 and belongs to the class of so-called Lipschitz networks, which were recently applied to defend against adversarial attacks. Moreover, due to its averaging property, PNNs can be used within so-called Plug-and-Play algorithms with convergence guarantee. In case of Parseval frames, we call the networks Parseval proximal neural networks (PPNNs). Then, the involved linear operators are in a Stiefel manifold and corresponding minimization methods can be applied for training of such networks. Finally, some proof-of-the concept examples demonstrate the performance of PPNNs.