{"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/tropical-geometry-of-deep-neural-networks","title":"Tropical Geometry of Deep Neural Networks","arxiv_id":"1805.07091","date":"2018-05-18","proceeding":"ICML 2018 7","authors":["Liwen Zhang","Gregory Naitzat","Lek-Heng Lim"],"abstract":"We establish, for the first time, connections between feedforward neural\nnetworks with ReLU activation and tropical geometry --- we show that the family\nof such neural networks is equivalent to the family of tropical rational maps.\nAmong other things, we deduce that feedforward ReLU neural networks with one\nhidden layer can be characterized by zonotopes, which serve as building blocks\nfor deeper networks; we relate decision boundaries of such neural networks to\ntropical hypersurfaces, a major object of study in tropical geometry; and we\nprove that linear regions of such neural networks correspond to vertices of\npolytopes associated with tropical rational functions. An insight from our\ntropical formulation is that a deeper network is exponentially more expressive\nthan a shallow network.","url_abs":"http://arxiv.org/abs/1805.07091v1","url_pdf":"http://arxiv.org/pdf/1805.07091v1.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":"tropical-geometry-of-deep-neural-networks","repo_url":"https://github.com/necoleman/relu_tropical_polynomial","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[{"method_slug":"relu","method_name":"ReLU"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1805.07091","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}