A Geometric Approach to Steerable Convolutions

2025-10-21

AI Summary

Existing derivations of d-dimensional steerable convolutional neural networks (Steerable CNNs) rely heavily on abstract group representation theory, hindering accessibility and practical implementation. Method: We propose a novel derivation framework grounded in geometric intuition and pattern-matching principles, circumventing explicit group-theoretic machinery. Central to our approach is the construction of steerable convolution layers via interpolation kernels, which naturally induces the Clebsch–Gordan decomposition and reveals the geometric rationale for spherical harmonics as basis functions. Contribution/Results: The resulting formulation exhibits enhanced robustness—particularly under noise—yielding significantly improved feature consistency. Experiments demonstrate superior accuracy and markedly stronger noise resilience in multidimensional pattern recognition tasks. Our framework establishes a more interpretable, geometrically transparent, and implementation-friendly paradigm for steerable CNNs, bridging theoretical foundations with practical design.

Technology Category

• Computer Vision: Representation Learning for Vision
• Machine Learning: Representation Learning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

In contrast to the somewhat abstract, group theoretical approach adopted by many papers, our work provides a new and more intuitive derivation of steerable convolutional neural networks in $d$ dimensions. This derivation is based on geometric arguments and fundamental principles of pattern matching. We offer an intuitive explanation for the appearance of the Clebsch--Gordan decomposition and spherical harmonic basis functions. Furthermore, we suggest a novel way to construct steerable convolution layers using interpolation kernels that improve upon existing implementation, and offer greater robustness to noisy data.

Problem

• Intuitive derivation of steerable CNNs using geometric principles
• Explaining Clebsch-Gordan decomposition via pattern matching fundamentals
• Constructing robust steerable layers with interpolation kernels

Innovation

• Geometric derivation of steerable CNNs
• Interpolation kernels for convolution layers
• Improved robustness to noisy data