【講座題目】PDO-eConvs: Partial Differential Operator Based Equivariant Convolutions
【主 講 人】林宙辰，北京大學教授
北京大學教授，IAPR/IEEE Fellow，國家杰青，中國圖象圖形學學會機器視覺專委會主任，中國自動化學會模式識別與機器智能專委會副主任。研究領域為計算機視覺、機器學習、圖像處理、模式識別和數值優化。發表論文200余篇，英文專著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 領域主席，IEEE T. PAMI、IJCV 編委。
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.