When learning from data, leveraging the symmetries of the domain the data lies on is a principled way to combat the curse of dimensionality: it constrains the set of functions to learn from. It is more data efficient than augmentation and gives a generalization guarantee. Symmetries might however be unknown or expensive to find; domains might not be homogeneous.

From the building blocks of space (vertices, edges, simplices), an incidence structure, and a metric---the domain's topology and geometry---a linear operator naturally emerges that commutes with any known and unknown symmetry action. We call that equivariant operator the generalized convolution operator. And we use it, designed or learned, to transform data and embed domains. In our generalized setting involving unknown and non-transitive symmetry groups, our convolution is an inner-product with a kernel that is localized instead of moved around by group actions like translations: a bias-variance tradeoff that paves the way to efficient learning on arbitrary discrete domains.

We develop convolutional neural networks that operate on graphs, meshes, and simplicial complexes. Their implementation amounts to the multiplications of data tensors by sparse matrices and pointwise operations, with linear compute, memory, and communication requirements. We demonstrate our method's efficiency by reaching state-of-the-art performance for multiple tasks on large discretizations of the sphere. DeepSphere has been used for studies in cosmology and shall be used for operational weather forecasting---advancing our understanding of the world and impacting billions of individuals.