Spherical Convolutionnal Neural Networks: Empirical Analysis of SCNNs

Convolutional neural networks (CNNs) are powerful tools in Deep Learning mainly due to their ability to exploit the translational symmetry present in images, as they are equivariant to translations. Other datasets present different types of symmetries (e.g. rotations), or lie on the sphere S2 (e.g. cosmological maps, omni-directional images, 3D models, ...). It is therefore of interest to design architectures that exploit the structure of the data and are equivariant to the 3D rotation group SO(3). Different architectures were designed to exploit these symmetries, such as 2D convolutions on planar projections, convolutions on the SO(3)group, or convolutions on graphs. The DeepSphere model approximates the sphere with a graph and performs graph convolutions. In this study, DeepSphere is evaluated against other spherical CNNs on different tasks.While the SO(3) convolution is equivariant to all rotations in SO(3), the graph convolutionis only equivariant to the rotations in S2 and invariant to the third rotation. Our experiments on SHREC-17 (a 3D shape retrieval task) show that DeepSphere achieves the same performance while being 40 times faster to train than Cohen et al. and 4 times faster than Esteves et al. Equivariance to the third rotation is an unnecessary price to pay. In order to prove these results, DeepSphere was tested on the similar dataset ModelNet40 (a shape classification task), and similar results as obtained by Esteves et al. were achieved. The odd behaviour with rotations (the performance worsens in presence of rotation perturbations) may be inherent to the task and the classes, instead of the models or the choice of the sampling scheme. Finally, regression tasks (both global and dense) were performed on GHCN-daily to prove the flexibility of DeepSphere with a non-hierarchical and irregular sampling of the sphere. The SCNN performed better than simply learning on the time series for each nodes.