Equivariant machine learning is an approach for designing deep learning models that respect the symmetries of the problem, with the aim of reducing model complexity and improving generalization. In this paper, we focus on an extension of shift equivariance, which is the basis of convolution networks on images, to general graphs. Unlike images, graphs do not have a natural notion of domain translation. Therefore, we consider the graph functional shifts as the symmetry group: the unitary operators that commute with the graph shift operator. Notably, such symmetries operate in the signal space rather than directly in the spatial space. We remark that each linear filter layer of a standard spectral graph neural network (GNN) commutes with graph functional shifts, but the activation function breaks this symmetry. Instead, we propose nonlinear spectral filters (NLSFs) that are fully equivariant to graph functional shifts and show that they have universal approximation properties. The proposed NLSFs are based on a new form of spectral domain that is transferable between graphs. We demonstrate the superior performance of NLSFs over existing spectral GNNs in node and graph classification benchmarks.
Scaling-Translation-Equivariant Networks with Decomposed Convolutional Filters
Department of Mathematics and Statistics, University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校数学与统计系）；Department of Electrical and Computer Engineering, Purdue University（普渡大学电气与计算机工程系）；Department of Mathematics, Duke University（杜克大学数学系）；Department of Electrical and Computer Engineering, Duke University（杜克大学电气与计算机工程系）
Equivariant Filters for Efficient Tracking in 3D Imaging
Brigham and Women’s Hospital（布莱根妇女医院）；Harvard Medical School（哈佛医学院）；CSAIL, Massachusetts Institute of Technology（麻省理工学院计算机科学与人工智能实验室）；Boston Children’s Hospital（波士顿儿童医院）
A rotation-equivariant convolutional neural network model of primary visual cortex
Center for Neuroscience and Artificial Intelligence, BCM（贝勒医学院神经科学与人工智能中心）；Baylor College of Medicine（贝勒医学院）；University of Tübingen（图宾根大学）；Bernstein Center for Computational Neuroscience, University of Tübingen（图宾根大学伯恩斯坦计算神经科学中心）；Max Planck Institute for Biological Cybernetics（马克斯·普朗克生物控制论研究所）；Rice University（莱斯大学）
An Efficient Conditional Score-based Filter for High Dimensional Nonlinear Filtering Problems
Department of Mathematical Sciences, Tsinghua University（清华大学数学科学系）；Yau Mathematical Sciences Center, Tsinghua University（清华大学叶中伟数学科学中心）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京燕琦湖数学科学与应用研究所）
Identifying Systems with Symmetries using Equivariant Autoregressive Reservoir Computers
Department of Statistics and Research, CNBS（统计与研究系，CNBS）；Department of Applied Mathematics, UNAH（应用数学系，UNAH）；Department of Chemical Engineering, Norwegian University of Science and Technology（化学工程系，挪威科学技术大学）
