Revisiting Spatial Invariance with Low-Rank Local Connectivity

Convolutional neural networks are among the most successful architectures in deep learning. This success is at least partially attributable to the efficacy of spatial invariance as an inductive bias. Locally connected layers, which differ from convolutional layers in their lack of spatial invariance, usually perform poorly in practice. However, these observations still leave open the possibility that some degree of relaxation of spatial invariance may yield a better inductive bias than either convolution or local connectivity. To test this hypothesis, we design a method to relax the spatial invariance of a network layer in a controlled manner. In particular, we create a \textit{low-rank} locally connected layer, where the filter bank applied at each position is constructed as a linear combination of basis set of filter banks. By varying the number of filter banks in the basis set, we can control the degree of departure from spatial invariance. In our experiments, we find that relaxing spatial invariance improves classification accuracy over both convolution and locally connected layers across MNIST, CIFAR-10, and CelebA datasets. These results suggest that spatial invariance in convolution layers may be overly restrictive.