Parseval Proximal Neural Networks

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.