Designing a convolution for a spherical neural network requires a delicate tradeoff between efficiency and rotation equivariance. DeepSphere, a method based on a graph representation of the discretized sphere, strikes a controllable balance between these two desiderata. This contribution is twofold. First, we study both theoretically and empirically how equivariance is affected by the underlying graph with respect to the number of pixels and neighbors. Second, we evaluate DeepSphere on relevant problems. Experiments show state-of-the-art performance and demonstrates the efficiency and flexibility of this formulation. Perhaps surprisingly, comparison with previous work suggests that anisotropic filters might be an unnecessary price to pay. Spherical data is found in many applications (figure 1). Planetary data (such as meteorological or geological measurements) and brain activity are example of intrinsically spherical data. The observation of the universe, LIDAR scans, and the digitalization of 3D objects are examples of projections due to observation. Labels or variables are often to be inferred from them. Examples are the inference of cosmological parameters from the distribution of mass in the universe , the segmentation of omnidirectional images (Khasanova & Frossard, 2017) , and the segmentation of cyclones from Earth observation (Mudigonda et al., 2017) . 2 A rigid full-sphere sampling is not ideal: brain activity is only measured on the scalp, the Milky Way's galactic plane masks observations, climate scientists desire a variable resolution, and the position of weather stations is arbitrary and changes over time. (e) Graphs can faithfully and efficiently represent sampled spherical data by placing vertices where it matters. As neural networks (NNs) have proved to be great tools for inference, variants have been developed to handle spherical data. Exploiting the locally Euclidean property of the sphere, early attempts used standard 2D convolutions on a grid sampling of the sphere (Boomsma & Frellsen, 2017; Su & Grauman, 2017; Coors et al., 2018) . While simple and efficient, those convolutions are not equivariant to rotations. On the other side of this tradeoff, Cohen et al. (2018) and Esteves et al. (2018) proposed to 2 METHOD DeepSphere leverages graph convolutions to achieve the following properties: (i) computational efficiency, (ii) sampling flexibility, and (iii) rotation equivariance (section 3). The main idea is to model the sampled sphere as a graph of connected pixels: the length of the shortest path between two pixels is an approximation of the geodesic distance between them. We use the graph CNN formulation introduced in (Defferrard et al., 2016 ) and a pooling strategy that exploits hierarchical samplings of the sphere. (ii) that a larger architecture can compensate for the lack of generality. We indeed observed that more feature maps and depth led to higher performance (section C.3). This work showed that DeepSphere strikes an interesting, and we think currently optimal, balance between desiderata for a spherical CNN. A single parameter, the number of neighbors k a pixel is connected to in the graph, controls the tradeoff between cost and equivariance (which is linked to performance). As computational cost and memory consumption scales linearly with the number of pixels, DeepSphere scales to spherical maps made of millions of pixels, a required resolution to faithfully represent cosmological and climate data. Also relevant in scientific applications is the flexibility offered by a graph representation (for partial coverage, missing data, and non-uniform samplings). Finally, the implementation of the graph convolution is straightforward, and the ubiquity of graph neural networks -pushing for their first-class support in DL frameworks -will make implementations even easier and more efficient. A potential drawback of graph Laplacian-based approaches is the isotropy of graph filters, reducing in principle the expressive power of the NN. Experiments from Cohen et al. (2019) and Boscaini et al. (2016) indeed suggest that more general convolutions achieve better performance. Our experiments on 3D shapes (section 4.1) and climate (section 4.3) however show that DeepSphere's isotropic filters do not hinder performance. Possible explanations for this discrepancy are that NNs somehow compensate for the lack of anisotropic filters, or that some tasks can be solved with isotropic filters. The distortions induced by the icosahedral projection in (Cohen et al., 2019) or the leakage of curvature information in (Boscaini et al., 2016) might also alter performance. Developing graph convolutions on irregular samplings that respect the geometry of the sphere is another research direction of importance. Practitioners currently interpolate their measurements (coming from arbitrarily positioned weather stations, satellites or telescopes) to regular samplings. This practice either results in a waste of resolution or computational and storage resources. Our ultimate goal is for practitioners to be able to work directly on their measurements, however distributed.
A graph-based spherical CNN that strikes an interesting balance of trade-offs for a wide variety of applications. Combines existing CNN frameworks based on the discretization of a sphere as a graph to show a convergence result which is related to the rotation equivalence on a sphere. The authors use the existing graph CNN formulation and a pooling strategy that exploits hierarchical pixelations of the sphere to learn from the discretized sphere.