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 $$\mathcal {H}$$ H and $$\mathcal {K}$$ K be real Hilbert spaces, $$b \in \mathcal {K}$$ b ∈ K and $$T \in \mathcal {B} (\mathcal {H},\mathcal {K})$$ T ∈ B ( H , K ) a linear operator with closed range and Moore–Penrose inverse $$T^\dagger $$ T † . Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator $$\mathrm {Prox}:\mathcal {K}\rightarrow \mathcal {K}$$ Prox : K → K the operator $$T^\dagger \, \mathrm {Prox}( T \cdot + b)$$ T † Prox ( T · + b ) is a proximity operator on $$\mathcal {H}$$ H equipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator $$\mathrm {Prox}= S_{\lambda }:\ell _2 \rightarrow \ell _2$$ Prox = S λ : ℓ 2 → ℓ 2 and any frame analysis operator $$T:\mathcal {H}\rightarrow \ell _2$$ T : H → ℓ 2 that the frame shrinkage operator $$T^\dagger \, S_\lambda \, T$$ T † S λ T is a proximity operator on a suitable Hilbert space. The concatenation of proximity operators on $$\mathbb R^d$$ 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.