Intrinsic and extrinsic deep learning on manifolds
This work addresses the challenge of applying deep learning to data on manifolds, which is important for fields like geometry and data analysis, but it appears incremental as it builds on existing manifold learning concepts.
The paper tackles the problem of deep learning on manifolds by proposing extrinsic and intrinsic deep neural network architectures, proving that their empirical risk minimizers converge at optimal rates, with iDNNs being accurate and fast-converging.
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.