Open Access
2024 Intrinsic and extrinsic deep learning on manifolds
Yihao Fang, Ilsang Ohn, Vijay Gupta, Lizhen Lin
Author Affiliations +
Electron. J. Statist. 18(1): 1160-1184 (2024). DOI: 10.1214/24-EJS2227


We propose extrinsic and intrinsic deep neural network architectures as general frameworks for deep learning on manifolds. Specifically, extrinsic deep neural networks (eDNNs) preserve geometric features on manifolds by utilizing an equivariant embedding from the manifold to its image in the Euclidean space. Moreover, intrinsic deep neural networks (iDNNs) incorporate the underlying intrinsic geometry of manifolds via exponential and log maps with respect to a Riemannian structure. Consequently, we prove that the empirical risk of the empirical risk minimizers (ERM) of eDNNs and iDNNs converge in optimal rates. Overall, The eDNNs framework is simple and easy to compute, while the iDNNs framework is accurate and fast converging. To demonstrate the utilities of our framework, various simulation studies, and real data analyses are presented with eDNNs and iDNNs.

Funding Statement

LL and YF are supported by grants DMS CAREER 1654579 and DMS 2113642. IO was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (NRF-2022R1F1A1069695) and Inha University Research Grant.


We would like to thank Dong Quan Nguyen, Steve Rosenberg, and Bayan Saparbayeva for very helpful discussions.


Download Citation

Yihao Fang. Ilsang Ohn. Vijay Gupta. Lizhen Lin. "Intrinsic and extrinsic deep learning on manifolds." Electron. J. Statist. 18 (1) 1160 - 1184, 2024.


Received: 1 February 2023; Published: 2024
First available in Project Euclid: 13 March 2024

Digital Object Identifier: 10.1214/24-EJS2227

Keywords: deep learning , eDNNs and iDNNs , Manifolds

Vol.18 • No. 1 • 2024
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