Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network (1806.09231v2)

Abstract: Recent work by Cohen \emph{et al.} has achieved state-of-the-art results for learning spherical images in a rotation invariant way by using ideas from group representation theory and noncommutative harmonic analysis. In this paper we propose a generalization of this work that generally exhibits improved performace, but from an implementation point of view is actually simpler. An unusual feature of the proposed architecture is that it uses the Clebsch--Gordan transform as its only source of nonlinearity, thus avoiding repeated forward and backward Fourier transforms. The underlying ideas of the paper generalize to constructing neural networks that are invariant to the action of other compact groups.

• The paper presents a novel spherical CNN architecture that uses the Clebsch-Gordan transform as its sole source of nonlinearity to operate entirely in Fourier space.
• It avoids the inefficiencies of back-and-forth transformations between real and Fourier domains, ensuring robust equivariance and computational simplicity.
• Empirical evaluations on spherical MNIST, molecular data, and 3D shape recognition tasks demonstrate state-of-the-art accuracy and practical versatility.

Clebsch--Gordan Nets: A Fully Fourier Space Spherical Convolutional Neural Network

The paper presents an innovative architectural approach to designing spherical convolutional neural networks (CNNs) with an emphasis on full Fourier space operations. The authors introduce the concept of Clebsch--Gordan Nets, which leverage the Clebsch--Gordan transform as the sole source of nonlinearity, thereby harnessing the simplicity of Fourier space computations and maintaining essential group invariance properties.

Overview and Technical Approach

The paper extends the work of previous research on spherical CNNs, which relied on group representation theory and noncommutative harmonic analysis to achieve rotation invariance in learning spherical images. The architecture introduced further improves performance while simplifying the implementation by avoiding the back-and-forth conversions between real space and Fourier space that were necessary in earlier models.

Key Components

1. Equivariance and Representation Theory: The foundation of this approach is the generalization of convolutional neural networks by adhering to the principle of equivariance to transformations. Specifically, the architecture utilizes noncommutative harmonic analysis to transform images on the sphere into a sequence of matrices representing spherical harmonics. This transformation guarantees that the network respects the natural rotational symmetries of data mapped on a sphere.
2. Clebsch--Gordan Transform: Unique to this paper, the Clebsch--Gordan transform replaces standard pointwise nonlinearities with a transformation that operates entirely in Fourier space. The idea is to decompose the Kronecker product of activation representations into a sum of irreducible representations using Clebsch--Gordan coefficients, preserving the equivariant properties through each layer of the network.
3. Efficient Implementation and Flexibility: By remaining exclusively in the domain of Fourier space, Clebsch--Gordan Nets avoid numerical inaccuracies and inefficiencies associated with spatial transformations, enabling not only rotation invariant processing on spherical domains but also providing a method that can apply to other compact group actions via analogous decompositions.

Numerical and Empirical Insights

The paper reports superior results for Clebsch--Gordan Nets across several spherical image tasks, such as rotationally transformed MNIST datasets adapted to spherical geometry, prediction tasks using molecular datasets, and $3D$ shape recognition. The architecture demonstrated robustness in handling roto-translations and maintained state-of-the-art accuracy compared to previous spherical CNN approaches. These numerical results bolster the assertion that using the Clebsch--Gordan transform for nonlinearity can lead to both computational advantages and performance improvements.

Implications and Future Developments

The discussion suggests that the ideas underlying Clebsch--Gordan Nets could be generalized beyond spherical CNNs to neural networks that seek invariance to various other group transformations, as long as these groups have a well-defined harmonic analysis framework. This advancement hints at future work focused on fully Fourier space architectures, exploring new group equivariances and potentially simplifying neural network designs in complex spatial domains. Future research may broaden the scope of applications ranging from physics simulations, computer graphics, to any domain where data naturally conforms to non-Euclidean symmetries.

In conclusion, Clebsch--Gordan Nets represent a significant methodological stride in designing neural networks that fully utilize the symmetry group algebra, specifically benefiting tasks involving spherical data. This research lays a foundation for exploring deeper theoretical insights and practical applications within the field of group-equivariant neural networks.