{"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/manifoldnet-a-deep-network-framework-for","title":"ManifoldNet: A Deep Network Framework for Manifold-valued Data","arxiv_id":"1809.06211","date":"2018-09-11","proceeding":null,"authors":["Rudrasis Chakraborty","Jose Bouza","Jonathan Manton","Baba C. Vemuri"],"abstract":"Deep neural networks have become the main work horse for many tasks involving\nlearning from data in a variety of applications in Science and Engineering.\nTraditionally, the input to these networks lie in a vector space and the\noperations employed within the network are well defined on vector-spaces. In\nthe recent past, due to technological advances in sensing, it has become\npossible to acquire manifold-valued data sets either directly or indirectly.\nExamples include but are not limited to data from omnidirectional cameras on\nautomobiles, drones etc., synthetic aperture radar imaging, diffusion magnetic\nresonance imaging, elastography and conductance imaging in the Medical Imaging\ndomain and others. Thus, there is need to generalize the deep neural networks\nto cope with input data that reside on curved manifolds where vector space\noperations are not naturally admissible. In this paper, we present a novel\ntheoretical framework to generalize the widely popular convolutional neural\nnetworks (CNNs) to high dimensional manifold-valued data inputs. We call these\nnetworks, ManifoldNets.\n  In ManifoldNets, convolution operation on data residing on Riemannian\nmanifolds is achieved via a provably convergent recursive computation of the\nweighted Fr\\'{e}chet Mean (wFM) of the given data, where the weights makeup the\nconvolution mask, to be learned. Further, we prove that the proposed wFM layer\nachieves a contraction mapping and hence ManifoldNet does not need the\nnon-linear ReLU unit used in standard CNNs. We present experiments, using the\nManifoldNet framework, to achieve dimensionality reduction by computing the\nprincipal linear subspaces that naturally reside on a Grassmannian. The\nexperimental results demonstrate the efficacy of ManifoldNets in the context of\nclassification and reconstruction accuracy.","url_abs":"http://arxiv.org/abs/1809.06211v3","url_pdf":"http://arxiv.org/pdf/1809.06211v3.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":"manifoldnet-a-deep-network-framework-for","repo_url":"https://github.com/jjbouza/manifold-net","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"dimensionality-reduction","task_name":"Dimensionality Reduction"}],"methods":[{"method_slug":"convolution","method_name":"Convolution"},{"method_slug":"relu","method_name":"ReLU"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1809.06211","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}