  No existing spherical convolutional neural network (CNN) framework is both
computationally scalable and rotationally equivariant. Continuous approaches
capture rotational equivariance but are often prohibitively computationally
demanding. Discrete approaches offer more favorable computational performance
but at the cost of equivariance. We develop a hybrid discrete-continuous
(DISCO) group convolution that is simultaneously equivariant and
computationally scalable to high-resolution. While our framework can be applied
to any compact group, we specialize to the sphere. Our DISCO spherical
convolutions exhibit $\text{SO}(3)$ rotational equivariance, where
$\text{SO}(n)$ is the special orthogonal group representing rotations in
$n$-dimensions. When restricting rotations of the convolution to the quotient
space $\text{SO}(3)/\text{SO}(2)$ for further computational enhancements, we
recover a form of asymptotic $\text{SO}(3)$ rotational equivariance. Through a
sparse tensor implementation we achieve linear scaling in number of pixels on
the sphere for both computational cost and memory usage. For 4k spherical
images we realize a saving of $10^9$ in computational cost and $10^4$ in memory
usage when compared to the most efficient alternative equivariant spherical
convolution. We apply the DISCO spherical CNN framework to a number of
benchmark dense-prediction problems on the sphere, such as semantic
segmentation and depth estimation, on all of which we achieve the
state-of-the-art performance.
