General $E(2)$-Equivariant Steerable CNNs (1911.08251v2)

Abstract: The big empirical success of group equivariant networks has led in recent years to the sprouting of a great variety of equivariant network architectures. A particular focus has thereby been on rotation and reflection equivariant CNNs for planar images. Here we give a general description of $E(2)$-equivariant convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs thereby yields constraints on the convolution kernels which depend on group representations describing the transformation laws of feature spaces. We show that these constraints for arbitrary group representations can be reduced to constraints under irreducible representations. A general solution of the kernel space constraint is given for arbitrary representations of the Euclidean group $E(2)$ and its subgroups. We implement a wide range of previously proposed and entirely new equivariant network architectures and extensively compare their performances. $E(2)$-steerable convolutions are further shown to yield remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as a drop-in replacement for non-equivariant convolutions.

• The paper introduces a novel strategy to solve convolution kernel constraints for E(2) using irreducible representations.
• It applies steerable feature fields to ensure rotation and reflection equivariance across diverse CNN architectures.
• Empirical benchmarks on datasets like MNIST and CIFAR show improved convergence, accuracy, and parameter efficiency over baseline models.

Overview of General E(2)-Equivariant Steerable CNNs

The paper presents a comprehensive framework for E(2)-equivariant steerable convolutional neural networks (CNNs), addressing the evolution of group-equivariant networks, with an emphasis on designs that guarantee rotation and reflection equivariance for planar images. This approach leverages the theory of steerable CNNs to systematically define and constrain convolution kernels based on group representations associated with feature spaces, and establishes a technique to manage these constraints by reducing them to irreducible representations.

Key Contributions

The central contribution of this paper is the development of a general strategy to solve kernel space constraints for arbitrary representations of the Euclidean group E(2) and its subgroups. The authors build on the steerable CNNs framework to propose a broad spectrum of equivariant models, accommodating both established and new architectural designs.

Methodology

The authors describe steerable feature fields as spaces that assign feature vectors with transformation laws defined by group representations. This ensures equivariant transformations under symmetry group actions. The constraints imposed on convolution kernels have been rigorously solved for E(2) by simplifying the problem to computations with irreducible representations.

• Irrep Decomposition: The reduction of convolutional kernel constraints to irreducible representations allows for efficient computation and practical implementation. This decomposition leads to simpler solutions and ensures computational feasibility even for larger representations.
• Kernel Solutions for Subgroups: The paper provides comprehensive tables and derivations for kernel spaces under various subgroup constraints, detailing solutions for the orthogonal group O(2) and its subgroups like the cyclic and dihedral groups.

Implementation and Results

The paper’s implementation spans various equivariant models, including regular group CNNs, Harmonic Networks, Vector Field Networks, and newly proposed architectures. An extensive benchmark paper is conducted on datasets like MNIST variations, CIFAR, and STL-10, demonstrating significant performance improvements over baseline CNN configurations. They highlight the practical benefits of these models in terms of faster convergence and enhanced parameter efficiency.

Implications and Future Directions

The implications of this work extend to various domains requiring rotational and reflectional equivariance, such as image recognition and biomedical imaging. The ability to adaptively restrict group actions according to data symmetries offers enhanced flexibility and may lead to more targeted architectures for specific tasks.

Future developments could focus on more complex manifold structures and expanding the scope of equivariant CNNs beyond planar images, potentially impacting broader areas in geometric deep learning and other applications where spatial symmetries play a critical role.

In summary, this paper lays robust theoretical and practical groundwork for designing and implementing E(2)-equivariant steerable CNNs, providing significant advancements in understanding and exploiting symmetries within neural network architectures.