Continuous Rotation Group Equivariant Network Inspired by Neural Population Coding

Neural population coding can represent continuous information by neurons with a series of discrete preferred stimuli, and we find that the bell-shaped tuning curve plays an important role in this mechanism. Inspired by this, we incorporate a bell-shaped tuning curve into the discrete group convolution to achieve continuous group equivariance. Simply, we modulate group convolution kernels by Gauss functions to obtain bell-shaped tuning curves. Benefiting from the modulation, kernels also gain smooth gradients on geometric dimensions (e.g., location dimension and orientation dimension). It allows us to generate group convolution kernels from sparse weights with learnable geometric parameters, which can achieve both competitive performances and parameter efficiencies. Furthermore, we quantitatively prove that discrete group convolutions with proper tuning curves (bigger than 1x sampling step) can achieve continuous equivariance. Experimental results show that 1) our approach achieves very competitive performances on MNIST-rot with at least 75% fewer parameters compared with previous SOTA methods, which is efficient in parameter; 2) Especially with small sample sizes, our approach exhibits more pronounced performance improvements (up to 24%); 3) It also has excellent rotation generalization ability on various datasets such as MNIST, CI-FAR, and ImageNet with both plain and ResNet architectures.