Building Transformation Layers for Riemannian Neural Networks

2026-09-28

AI Summary

This study addresses the limited generalizability of existing manifold neural networks, which typically rely on specific geometric properties. We propose a unified computational framework for fully connected and convolutional layers in Riemannian spaces, integrating ten representative manifolds—including hyperbolic, symmetric positive definite (SPD), and Grassmannian manifolds—into a generalized paradigm that enables cross-manifold network layer construction. Grounded in Riemannian geometry theory, this approach overcomes the limitations of single-geometry modeling by providing a principled formulation applicable across diverse geometric structures. Extensive experiments conducted on ten canonical manifolds validate both the effectiveness and broad applicability of the proposed method. The source code has been made publicly available to facilitate reproducibility and further research.

Abstract

Recently, deep neural networks on manifold-valued representations have garnered significant attention across various machine learning applications. One recent focus is the generalization of Euclidean fully connected (FC) and convolutional layers to non-Euclidean geometries. However, previous approaches typically focus on a few selected manifolds and rely on specific properties of the target manifold. In contrast, this work proposes a framework for constructing FC and convolutional layers over computationally tractable Riemannian spaces. This framework incorporates several previous FC layers across different geometries as special cases and is instantiated on ten representative manifolds, including three hyperbolic models, five geometries of the symmetric positive definite (SPD) manifold, and two Grassmannian perspectives. Experiments on different manifolds demonstrate the effectiveness and applicability of our approach. Code can be found at https://github.com/GitZH-Chen/RieTrans.

Problem

• Riemannian Neural Networks
• Manifold-valued Representations
• Fully Connected Layers
• Convolutional Layers
• Non-Euclidean Geometries

Innovation

• Riemannian Neural Networks
• Transformation Layers
• Fully Connected Layers
• Convolutional Layers
• Manifold-valued Representations