Framework builds neural network layers on diverse curved spaces
Building Transformation Layers for Riemannian Neural Networks
Artificial IntelligenceMachine Learning
Summary
Some data naturally lives on curved spaces rather than flat ones, which makes standard neural networks less effective. The authors present a new way to create neural network layers that work on many types of curved spaces, known as Riemannian manifolds. Their design includes and generalizes previous methods that were limited to special cases. They tested their approach on multiple curved spaces and showed it works well.
What this means in practice
- •For machine learning engineers: Design neural networks capable of handling data represented in various non-flat geometric spaces for better model flexibility.
- •For computer vision developers: Build image analysis tools that use advanced layers to handle manifold-valued features such as SPD matrices from covariance descriptors.
Authors
Ziheng Chen
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.