Riemannian Deep Learning:Modules, Networks, and Geometries

Imagine you're developing machine learning models that need to work with complex data structures, like those found in geometry or physics. Traditional approaches often rely on Euclidean spaces, which can lead to inaccuracies when dealing with data that naturally resides on curved surfaces or other non-Euclidean spaces. This can result in models that are not only less effective but also computationally expensive and prone to numerical instability, especially when performing operations that require precise geometric calculations. This is what's called the limitation of relying on Euclidean approximations in manifold-valued representations.

To address these challenges, the paper proposes a comprehensive framework for Riemannian deep learning that encompasses reusable neural modules, specialized network architectures, and innovative geometric designs. By generalizing techniques like batch normalization and multinomial logistic regression to work across a wider range of geometric spaces, including Lie groups and SPD manifolds, the framework allows for more robust and efficient learning. Additionally, it introduces adaptive Riemannian metrics that enhance computational efficiency and stability. Compared to prior work, this approach not only broadens the applicability of deep learning to more complex data structures but also improves the performance and reliability of models in fields like vision, signal processing, and genomics.

The framework introduces significant generalizations for Riemannian deep learning.

The methods are supported by theoretical analysis and numerical experiments across various applications.