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Parseval Proximal Neural Networks
Marzieh Hasannasab, Johannes Hertrich, Sebastian Neumayer, Gerlind Plonka, Simon Setzer, Gabriele Steidl
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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) have 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 Prox𝒦→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_λℓ₂ →ℓ₂ and any frame analysis operator Tℋ→ℓ₂ that the frame shrinkage operator T^† S_λ T is a proximity operator on a suitable Hilbert space. The concatenation of proximity operators on ℝᵈ equipped with different norms establishes a PNN. If the network arises from tight frame analysis or synthesis operators, then it forms an averaged operator. Hence, 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. Finally, some proof-of-the concept examples demonstrate the performance of PPNNs.
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