{"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/topological-grammars-for-data-approximation","title":"Topological Grammars for Data Approximation","arxiv_id":"cs/0603090","date":"2006-03-22","proceeding":null,"authors":["A. N. Gorban","N. R. Sumner","A. Y. Zinovyev"],"abstract":"A method of {\\it topological grammars} is proposed for multidimensional data approximation. For data with complex topology we define a {\\it principal cubic complex} of low dimension and given complexity that gives the best approximation for the dataset. This complex is a generalization of linear and non-linear principal manifolds and includes them as particular cases. The problem of optimal principal complex construction is transformed into a series of minimization problems for quadratic functionals. These quadratic functionals have a physically transparent interpretation in terms of elastic energy. For the energy computation, the whole complex is represented as a system of nodes and springs. Topologically, the principal complex is a product of one-dimensional continuums (represented by graphs), and the grammars describe how these continuums transform during the process of optimal complex construction. This factorization of the whole process onto one-dimensional transformations using minimization of quadratic energy functionals allow us to construct efficient algorithms.","url_abs":"https://arxiv.org/abs/cs/0603090v2","url_pdf":"https://arxiv.org/pdf/cs/0603090v2.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":"topological-grammars-for-data-approximation","repo_url":"https://github.com/auranic/Elastic-principal-graphs","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"topological-grammars-for-data-approximation","repo_url":"https://github.com/sysbio-curie/ElPiGraph.M","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}