{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/parseval-proximal-neural-networks","title":"Parseval Proximal Neural Networks","arxiv_id":"1912.10480","date":"2019-12-19","proceeding":null,"authors":["Marzieh Hasannasab","Johannes Hertrich","Sebastian Neumayer","Gerlind Plonka","Simon Setzer","Gabriele Steidl"],"abstract":"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 $\\mathcal H$ and $\\mathcal K$ be real Hilbert spaces, $b\\in\\mathcal K$ and $T\\in\\mathcal{B}(\\mathcal H,\\mathcal K)$ have closed range and Moore-Penrose inverse $T^\\dagger$. Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator $\\text{Prox}\\colon\\mathcal K\\to\\mathcal K$ the operator $T^\\dagger\\,\\text{Prox} (T\\cdot +b)$ is a proximity operator on $\\mathcal H$ equipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator $\\text{Prox} = S_{\\lambda}\\colon\\ell_2 \\rightarrow\\ell_2$ and any frame analysis operator $T\\colon\\mathcal H\\to\\ell_2$ that the frame shrinkage operator $T^\\dagger\\, S_\\lambda\\,T$ is a proximity operator on a suitable Hilbert space. The concatenation of proximity operators on $\\mathbb R^d$ equipped with different norms establishes a PNN. If the network arises from tight frame analysis or synthesis operators, then it forms an averaged operator. 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.","url_abs":"http://arxiv.org/abs/1912.10480v2","url_pdf":"http://arxiv.org/pdf/1912.10480v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"parseval-proximal-neural-networks","repo_url":"https://github.com/johertrich/Proximal_Neural_Networks","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1912.10480","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}