Intrinsic and extrinsic deep learning on manifolds

TL;DR

This paper introduces intrinsic and extrinsic deep neural network architectures tailored for learning on manifolds, leveraging geometric properties and Riemannian structures to improve accuracy and convergence.

Contribution

It develops general frameworks for deep learning on manifolds using intrinsic and extrinsic approaches, with proven convergence rates and practical algorithms.

Findings

eDNNs are simple and computationally efficient.

iDNNs are highly accurate and converge quickly.

Frameworks are validated through simulations and real data.

Abstract

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.