{"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/intertwiners-between-induced-representations","title":"Intertwiners between Induced Representations (with Applications to the Theory of Equivariant Neural Networks)","arxiv_id":"1803.10743","date":"2018-03-28","proceeding":null,"authors":["Taco S. Cohen","Mario Geiger","Maurice Weiler"],"abstract":"Group equivariant and steerable convolutional neural networks (regular and\nsteerable G-CNNs) have recently emerged as a very effective model class for\nlearning from signal data such as 2D and 3D images, video, and other data where\nsymmetries are present. In geometrical terms, regular G-CNNs represent data in\nterms of scalar fields (\"feature channels\"), whereas the steerable G-CNN can\nalso use vector or tensor fields (\"capsules\") to represent data. In algebraic\nterms, the feature spaces in regular G-CNNs transform according to a regular\nrepresentation of the group G, whereas the feature spaces in Steerable G-CNNs\ntransform according to the more general induced representations of G. In order\nto make the network equivariant, each layer in a G-CNN is required to\nintertwine between the induced representations associated with its input and\noutput space.\n  In this paper we present a general mathematical framework for G-CNNs on\nhomogeneous spaces like Euclidean space or the sphere. We show, using\nelementary methods, that the layers of an equivariant network are convolutional\nif and only if the input and output feature spaces transform according to an\ninduced representation. This result, which follows from G.W. Mackey's abstract\ntheory on induced representations, establishes G-CNNs as a universal class of\nequivariant network architectures, and generalizes the important recent work of\nKondor & Trivedi on the intertwiners between regular representations.","url_abs":"http://arxiv.org/abs/1803.10743v2","url_pdf":"http://arxiv.org/pdf/1803.10743v2.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":"intertwiners-between-induced-representations","repo_url":"https://github.com/jonkhler/s2cnn","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"unanswered"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1803.10743","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}