[Paper Review] DeepSphere: a graph-based spherical CNN

Michaël Defferrard, Martino Milani+2 more|arXiv (Cornell University)|Dec 30, 2020

3D Shape Modeling and Analysis21 references17 citations

TL;DR

DeepSphere proposes a graph-based spherical CNN that balances computational efficiency and rotation equivariance by modeling the sphere as a weighted graph with controllable neighborhood connectivity. Experiments show state-of-the-art performance on climate, cosmological, and 3D shape tasks, demonstrating that isotropic graph filters can be sufficient despite theoretical advantages of anisotropic ones.

ABSTRACT

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 sampled 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 vertices 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. Our code is available at https://github.com/deepsphere

Motivation & Objective

• Address the trade-off between computational efficiency and rotation equivariance in spherical CNNs for scientific applications.
• Enable neural networks to operate directly on irregular, non-uniformly sampled spherical data such as weather stations or brain recordings.
• Develop a method that maintains high performance while scaling to large-scale datasets with millions of points.
• Investigate whether isotropic graph filters—common in graph neural networks—can match or exceed the performance of more complex, anisotropic spherical convolutions.
• Provide a flexible, scalable framework that avoids the need for interpolation or projection to regular grids, preserving data fidelity.

Proposed method

• Represent the sphere as a weighted undirected graph where vertices correspond to sampled points and edge weights reflect geodesic distance-based similarity.
• Construct the graph Laplacian $\mathbf{L} = \mathbf{D} - \mathbf{A}$ using a kernelized distance measure with $k$-nearest neighbors and adaptive width $t$.
• Apply spectral graph convolutions via polynomial approximation $h(\mathbf{L})\mathbf{f} = \left(\sum_{i=0}^{P} \alpha_i \mathbf{L}^i\right)\mathbf{f}$, enabling efficient, localized message passing.
• Use hierarchical sampling and pooling to maintain spatial resolution and enable multi-scale analysis on irregular grids.
• Train the model end-to-end using RMSprop with MSE loss on tasks like temperature prediction and day-of-year regression from climate data.
• Control equivariance via the number of neighbors $k$, allowing a tunable trade-off between performance and computational cost.

Experimental results

Research questions

• RQ1 How does the number of neighbors $k$ in the graph construction affect rotation equivariance and model performance?
• RQ2 Can isotropic graph filters achieve state-of-the-art performance on spherical learning tasks despite theoretical advantages of anisotropic filters?
• RQ3 To what extent does non-uniform sampling (e.g., weather stations) degrade equivariance, and can graph-based methods still outperform regularized or projected alternatives?
• RQ4 How does DeepSphere compare to spherical harmonic-based and icosahedral projection-based methods in terms of accuracy and scalability?
• RQ5 Can graph-based convolutions on irregular spherical samplings eliminate the need for data interpolation or projection to regular grids?

Key findings

• Using polynomial order $P=4$ instead of $P=0$ reduced MSE from 10.88 to 8.20 in dense temperature regression, showing that neighborhood aggregation improves performance.
• For day-of-year prediction from temperature, R² improved from 0.882 to 0.969 when using $P=4$, demonstrating the benefit of spatial inductive bias.
• On precipitation-based day prediction, $P=4$ achieved R² = 0.597 compared to -0.980 with $P=0$, indicating that spatial structure is critical for complex patterns.
• Despite lacking anisotropic filters, DeepSphere achieved state-of-the-art performance on 3D shape and climate data, challenging assumptions about filter isotropy.
• The method scales linearly with the number of vertices, enabling efficient training on datasets with up to ~10,000 weather stations.
• The graph-based approach maintained strong performance even on highly non-uniform samplings (e.g., North America-dense weather stations), outperforming regularized or projected alternatives in key benchmarks.