PDO-eConvs: Partial Differential Operator Based


【讲座题目】PDO-eConvs: Partial Differential Operator Based Equivariant Convolutions



【主 人】林宙辰,北京大学教授


北京大学教授,IAPR/IEEE Fellow,国家杰青,中国图象图形学学会机器视觉专委会主任,中国自动化学会模式识别与机器智能专委会副主任。研究领域为计算机视觉、机器学习、图像处理、模式识别和数值优化。发表论文200余篇,英文专著2本。任 CVPR 2014/2016/2019/2020/2021 、 ICCV 2015 、 NIPS 2015/2018/2019/2020、ICML 2020、IJCAI 2020/2021、AAAI 2019/2020 和 ICLR 2021 领域主席,IEEE T. PAMI、IJCV 编委。


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. 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.