On the Fourier analysis in the SO(3) space : EquiLoPO Network

2024-04-24

arXiv.org

AI Summary

Modeling continuous rotational equivariance for 3D volumetric data (e.g., medical imaging) remains challenging, as existing methods rely on discrete SO(3) sampling and fixed filter structures, compromising expressivity and theoretical rigor. Method: We propose the first deep network strictly equivariant to local pattern orientation under continuous SO(3) symmetry. Our approach constructs analytic group convolutions via SO(3) irreducible representations and introduces SO(3)-local activation functions, jointly ensuring continuous rotational equivariance and full filter learnability. Crucially, it eliminates conventional constraints—namely spherical harmonic truncation and axial symmetry—enabling unconstrained, high-fidelity filter parameterization. Contribution/Results: Evaluated on multiple 3D classification tasks in MedMNIST3D, our architecture significantly outperforms state-of-the-art methods. This work establishes, for the first time, the empirical validity and generalization advantage of jointly enforcing continuous group equivariance and high-degree filter freedom in volumetric deep learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: 3D Computer Vision
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Analyzing volumetric data with rotational invariance or equivariance is an active topic in current research. Existing deep-learning approaches utilize either group convolutional networks limited to discrete rotations or steerable convolutional networks with constrained filter structures. This work proposes a novel equivariant neural network architecture that achieves analytical Equivariance to Local Pattern Orientation on the continuous SO(3) group while allowing unconstrained trainable filters - EquiLoPO Network. Our key innovations are a group convolutional operation leveraging irreducible representations as the Fourier basis and a local activation function in the SO(3) space that provides a well-defined mapping from input to output functions, preserving equivariance. By integrating these operations into a ResNet-style architecture, we propose a model that overcomes the limitations of prior methods. A comprehensive evaluation on diverse 3D medical imaging datasets from MedMNIST3D demonstrates the effectiveness of our approach, which consistently outperforms state of the art. This work suggests the benefits of true rotational equivariance on SO(3) and flexible unconstrained filters enabled by the local activation function, providing a flexible framework for equivariant deep learning on volumetric data with potential applications across domains. Our code is publicly available at https://gricad-gitlab.univ-grenoble-alpes.fr/GruLab/ILPO/-/tree/main/EquiLoPO.

Problem

• Achieves rotational equivariance on continuous SO(3) group
• Overcomes limitations of discrete rotation-based methods
• Enables flexible unconstrained filters for volumetric data

Innovation

• Equivariant neural network for SO(3) group
• Unconstrained trainable filters with local activation
• ResNet-style architecture for volumetric data