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Convolutional Neural Networks (CNNs) have become the method of choice forlearning problems involving 2D planar images. However, a number of problems ofrecent interest have created a demand for models that can analyze sphericalimages. Examples include omnidirectional vision for drones, robots, andautonomous cars, molecular regression problems, and global weather and climatemodelling. A naive application of convolutional networks to a planar projectionof the spherical signal is destined to fail, because the space-varyingdistortions introduced by such a projection will make translational weightsharing ineffective.In this paper we introduce the building blocks for constructing sphericalCNNs. We propose a definition for the spherical cross-correlation that is bothexpressive and rotation-equivariant. The spherical correlation satisfies ageneralized Fourier theorem, which allows us to compute it efficiently using ageneralized (non-commutative) Fast Fourier Transform (FFT) algorithm. Wedemonstrate the computational efficiency, numerical accuracy, and effectivenessof spherical CNNs applied to 3D model recognition and atomization energyregression.