Scalable and Equivariant Spherical CNNs by Discrete-Continuous (DISCO) Convolutions

TL;DR

This paper introduces DISCO, a hybrid discrete-continuous spherical CNN framework that achieves rotational equivariance and high computational scalability, enabling state-of-the-art performance on spherical dense-prediction tasks.

Contribution

The paper develops a novel hybrid discrete-continuous group convolution for spherical CNNs that is both equivariant and scalable, addressing limitations of prior approaches.

Findings

Achieves linear scaling in pixels for cost and memory.

Realizes significant computational and memory savings over existing methods.

Sets new state-of-the-art results on spherical dense-prediction benchmarks.

Abstract

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.