Nonlinearities in Steerable SO(2)-Equivariant CNNs
This addresses a key architectural limitation in equivariant neural networks for researchers and practitioners in fields like computer vision and geometric deep learning, offering a novel method to incorporate nonlinearities without sacrificing symmetry, though it is incremental in advancing existing steerable CNN frameworks.
The paper tackles the problem of maintaining equivariance in steerable CNNs when using nonlinear layers, which typically break symmetry, by applying harmonic distortion analysis to Fourier representations of SO(2) and developing an FFT-based algorithm for exact or approximate equivariance with polynomial or general nonlinearities. It results in a fully E(3)-equivariant network for 3D surface data that achieves competitive accuracy compared to state-of-the-art methods while enabling continuous symmetry and exact equivariance.
Invariance under symmetry is an important problem in machine learning. Our paper looks specifically at equivariant neural networks where transformations of inputs yield homomorphic transformations of outputs. Here, steerable CNNs have emerged as the standard solution. An inherent problem of steerable representations is that general nonlinear layers break equivariance, thus restricting architectural choices. Our paper applies harmonic distortion analysis to illuminate the effect of nonlinearities on Fourier representations of SO(2). We develop a novel FFT-based algorithm for computing representations of non-linearly transformed activations while maintaining band-limitation. It yields exact equivariance for polynomial (approximations of) nonlinearities, as well as approximate solutions with tunable accuracy for general functions. We apply the approach to build a fully E(3)-equivariant network for sampled 3D surface data. In experiments with 2D and 3D data, we obtain results that compare favorably to the state-of-the-art in terms of accuracy while permitting continuous symmetry and exact equivariance.