PDO-eConvs: Partial Differential Operator Based Equivariant Convolutions
This work addresses the problem of improving equivariance in neural networks for computer vision tasks, offering a more parameter-efficient method, though it is incremental as it builds on existing equivariance research.
The authors tackled the limited equivariance of neural networks on discrete grids by connecting convolutions to partial differential operators, proposing PDO-eConvs that achieve approximate equivariance to the n-dimensional Euclidean group with quadratic-order error. Experiments on rotated MNIST and natural image classification show competitive performance with 12.6% of the parameters compared to Wide ResNets.
Recent research has shown that incorporating equivariance into neural network architectures is very helpful, and there have been some works investigating the equivariance of networks under group actions. However, as digital images and feature maps are on the discrete meshgrid, corresponding equivariance-preserving transformation groups are very limited. In this work, we deal with this issue from the connection between convolutions and partial differential operators (PDOs). In theory, assuming inputs to be smooth, we transform PDOs and propose a system which is equivariant to a much more general continuous group, the $n$-dimension Euclidean group. In implementation, we discretize the system using the numerical schemes of PDOs, deriving approximately equivariant convolutions (PDO-eConvs). Theoretically, the approximation error of PDO-eConvs is of the quadratic order. It is the first time that the error analysis is provided when the equivariance is approximate. Extensive experiments on rotated MNIST and natural image classification show that PDO-eConvs perform competitively yet use parameters much more efficiently. Particularly, compared with Wide ResNets, our methods result in better results using only 12.6% parameters.