{"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/towards-a-topological-geometrical-theory-of","title":"Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning","arxiv_id":"1812.11832","date":"2018-12-31","proceeding":null,"authors":["Mattia G. Bergomi","Patrizio Frosini","Daniela Giorgi","Nicola Quercioli"],"abstract":"The aim of this paper is to provide a general mathematical framework for\ngroup equivariance in the machine learning context. The framework builds on a\nsynergy between persistent homology and the theory of group actions. We define\ngroup-equivariant non-expansive operators (GENEOs), which are maps between\nfunction spaces associated with groups of transformations. We study the\ntopological and metric properties of the space of GENEOs to evaluate their\napproximating power and set the basis for general strategies to initialise and\ncompose operators. We begin by defining suitable pseudo-metrics for the\nfunction spaces, the equivariance groups, and the set of non-expansive\noperators. Basing on these pseudo-metrics, we prove that the space of GENEOs is\ncompact and convex, under the assumption that the function spaces are compact\nand convex. These results provide fundamental guarantees in a machine learning\nperspective. We show examples on the MNIST and fashion-MNIST datasets. By\nconsidering isometry-equivariant non-expansive operators, we describe a simple\nstrategy to select and sample operators, and show how the selected and sampled\noperators can be used to perform both classical metric learning and an\neffective initialisation of the kernels of a convolutional neural network.","url_abs":"http://arxiv.org/abs/1812.11832v3","url_pdf":"http://arxiv.org/pdf/1812.11832v3.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":"towards-a-topological-geometrical-theory-of","repo_url":"https://gitlab.com/mattia.bergomi/geneos","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"machine-learning","task_name":"BIG-bench Machine Learning"},{"task_slug":"metric-learning","task_name":"Metric Learning"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1812.11832","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}