{"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/input-output-equivalence-of-unitary-and-1","title":"Input-Output Equivalence of Unitary and Contractive RNNs","arxiv_id":null,"date":"2019-12-01","proceeding":"NeurIPS 2019 12","authors":["Melikasadat Emami","Mojtaba Sahraee Ardakan","Sundeep Rangan","Alyson K. Fletcher"],"abstract":"Unitary recurrent neural networks (URNNs) have been proposed as a method to overcome the vanishing and exploding gradient problem in modeling data with long-term dependencies. A basic question is how restrictive is the unitary constraint on the possible input-output mappings of such a network? This works shows that for any contractive RNN with ReLU activations, there is a URNN with at most twice the number of hidden states and the identical input-output mapping.  Hence, with ReLU activations, URNNs are as expressive as general RNNs.  In contrast, for certain smooth activations, it is shown that the input-output mapping of an RNN cannot be matched with a URNN, even with an arbitrary number of states.  The theoretical results are supported by experiments on modeling of slowly-varying dynamical systems.","url_abs":"http://papers.nips.cc/paper/9671-input-output-equivalence-of-unitary-and-contractive-rnns","url_pdf":"http://papers.nips.cc/paper/9671-input-output-equivalence-of-unitary-and-contractive-rnns.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":"abstracts"},"code_links":[{"paper_slug":"input-output-equivalence-of-unitary-and-1","repo_url":"https://github.com/melikaemami/URNN","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"tf","reach":null}],"tasks":[],"methods":[{"method_slug":"relu","method_name":"ReLU"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}