{"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/learning-unitary-operators-with-help-from-un","title":"Learning Unitary Operators with Help From u(n)","arxiv_id":"1607.04903","date":"2016-07-17","proceeding":null,"authors":["Stephanie L. Hyland","Gunnar Rätsch"],"abstract":"A major challenge in the training of recurrent neural networks is the\nso-called vanishing or exploding gradient problem. The use of a norm-preserving\ntransition operator can address this issue, but parametrization is challenging.\nIn this work we focus on unitary operators and describe a parametrization using\nthe Lie algebra $\\mathfrak{u}(n)$ associated with the Lie group $U(n)$ of $n\n\\times n$ unitary matrices. The exponential map provides a correspondence\nbetween these spaces, and allows us to define a unitary matrix using $n^2$ real\ncoefficients relative to a basis of the Lie algebra. The parametrization is\nclosed under additive updates of these coefficients, and thus provides a simple\nspace in which to do gradient descent. We demonstrate the effectiveness of this\nparametrization on the problem of learning arbitrary unitary operators,\ncomparing to several baselines and outperforming a recently-proposed\nlower-dimensional parametrization. We additionally use our parametrization to\ngeneralize a recently-proposed unitary recurrent neural network to arbitrary\nunitary matrices, using it to solve standard long-memory tasks.","url_abs":"http://arxiv.org/abs/1607.04903v3","url_pdf":"http://arxiv.org/pdf/1607.04903v3.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":"learning-unitary-operators-with-help-from-un","repo_url":"https://github.com/ratschlab/uRNN","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1607.04903","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}