{"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/on-transversality-of-bent-hyperplane","title":"On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networks","arxiv_id":"2008.09052","date":"2020-08-20","proceeding":null,"authors":["J. Elisenda Grigsby","Kathryn Lindsey"],"abstract":"Let F:R^n -> R be a feedforward ReLU neural network. It is well-known that for any choice of parameters, F is continuous and piecewise (affine) linear. We lay some foundations for a systematic investigation of how the architecture of F impacts the geometry and topology of its possible decision regions for binary classification tasks. Following the classical progression for smooth functions in differential topology, we first define the notion of a generic, transversal ReLU neural network and show that almost all ReLU networks are generic and transversal. We then define a partially-oriented linear 1-complex in the domain of F and identify properties of this complex that yield an obstruction to the existence of bounded connected components of a decision region. We use this obstruction to prove that a decision region of a generic, transversal ReLU network F: R^n -> R with a single hidden layer of dimension (n + 1) can have no more than one bounded connected component.","url_abs":"https://arxiv.org/abs/2008.09052v2","url_pdf":"https://arxiv.org/pdf/2008.09052v2.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":"on-transversality-of-bent-hyperplane","repo_url":"https://github.com/mmasden/canonicalpoly","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"binary-classification","task_name":"Binary Classification"}],"methods":[{"method_slug":"relu","method_name":"ReLU"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2008.09052","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}