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3D连续变换的等变性模型构建",{"type":18,"tag":26,"props":314,"children":315},{},[316],{"type":24,"value":317},"在很多科学问题中，我们关注的是3D空间中连续的旋转和平移对称性，例如化学分子的结构发生旋转和平移，预测的分子属性构成的向量将发生对应的变换。这些连续的旋转变换R和平移变换t组成SE(3)群中的元素，并且这些变换可以表示为向量空间中的变换矩阵。不同的向量空间中的变换矩阵可能不同，但这些向量空间都可以分解为相互独立的子向量空间。每个子空间内有相同的变换规律，即群中所有的变换元素作用到子空间的向量上得到的向量还在该子空间内，因此群中的变换元素可以用该子空间上不可约简的变换矩阵表示。例如，总能量、能隙等标量在SE(3)群元素的作用下保持不变，其变换矩阵表示为D^0(R)=1；力场等3D向量下SE(3)群元素的作用下发生相应的旋转，其变换矩阵表示为D^1(R)=R；在更高维的向量空间中，D^l(R)是2l+1维的方阵。这些变换矩阵D^l(R)称为旋转R对应的l阶Wigner-D矩阵，而对应的子向量空间成为SE(3)群的l阶不可约不变子空间，其中的向量称为l阶等变向量。而在平移变换下，这些向量总是保持不变，因为我们关心的性质只与相对位置有关。",{"type":18,"tag":26,"props":319,"children":320},{},[321],{"type":24,"value":322},"通常把3D几何信息映射到SE(3)群的不变子空间中的特征的方法是采用球谐函数映射。球谐函数Y^l将一个3维向量映射成一个2l+1维向量，其代表输入向量分解成2l+1个基球谐函数时的系数。如下图所示，由于只使用了有限数量的基，该三维向量代表的球面上的delta函数会有一定的展宽。",{"type":18,"tag":26,"props":324,"children":325},{},[326],{"type":18,"tag":92,"props":327,"children":329},{"alt":94,"src":328},"https://fileserver.developer.huaweicloud.com/FileServer/getFile/cmtybbs/e64/154/b38/90a1d5d431e64154b387b3660e356ff5.20230913074648.28725684044160357487109482342249:50540913012303:2400:BAB0A4B987ED4AAA20AD4A09F4EBD56532B326FA3F448206A8BF585FF975A2AA.png",[],{"type":18,"tag":26,"props":331,"children":332},{},[333],{"type":18,"tag":92,"props":334,"children":337},{"alt":335,"src":336},"cke_8222.png","https://fileserver.developer.huaweicloud.com/FileServer/getFile/cmtybbs/e64/154/b38/90a1d5d431e64154b387b3660e356ff5.20230913074703.48030212109407735978885537300231:50540913012303:2400:A4B05E5CEB80D3F2DD2744F5E4FF2540173AD26572797CEF8162B839CC712C36.png",[],{"type":18,"tag":26,"props":339,"children":340},{},[341],{"type":18,"tag":92,"props":342,"children":344},{"alt":94,"src":343},"https://fileserver.developer.huaweicloud.com/FileServer/getFile/cmtybbs/e64/154/b38/90a1d5d431e64154b387b3660e356ff5.20230913074712.08946610252587761072878406171574:50540913012303:2400:78FC9519249B2EB302E9D14CAA0467FAFC9706D1D248BA9070E5A5FDEFE930DD.png",[],{"type":18,"tag":26,"props":346,"children":347},{},[348],{"type":18,"tag":92,"props":349,"children":352},{"alt":350,"src":351},"cke_11824.png","https://fileserver.developer.huaweicloud.com/FileServer/getFile/cmtybbs/e64/154/b38/90a1d5d431e64154b387b3660e356ff5.20230913074731.91870662528877683047681085243289:50540913012303:2400:69F150495968CFA1DA32FC911E245F8F7F8F32DCA8F4E90F22FD36C76690AC5E.png",[],{"type":18,"tag":26,"props":354,"children":355},{},[356],{"type":24,"value":357},"其中g是空间变换群中的变换，ρ_in、ρ_out分别代表该变换在输入和输出特征空间中的表示（即转换矩阵）。",{"type":18,"tag":26,"props":359,"children":360},{},[361],{"type":24,"value":362},"至此，文章对对称性和等变性的理论阐述基本结束，后面即是对第一章中列出的多个领域的分别概述。",{"type":18,"tag":364,"props":365,"children":367},"h2",{"id":366},"参考文献",[368],{"type":18,"tag":32,"props":369,"children":370},{},[371],{"type":24,"value":366},{"type":18,"tag":26,"props":373,"children":374},{},[375],{"type":24,"value":376},"[1] Ren P, Rao C, Liu Y, et al. 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Group Equivariant Convolutional Networks. In International Conference on Machine Learning. PMLR, 48:2990–2999.",{"type":18,"tag":26,"props":407,"children":408},{},[409,411],{"type":24,"value":410},"【3】 Nathaniel Thomas, Tess Smidt, Steven Kearnes, et al. 2018. Tensor field networks: Rotation-and translation-equivariant neural networks for 3d point clouds. arXiv: ",{"type":18,"tag":383,"props":412,"children":415},{"href":413,"rel":414},"https://arxiv.org/abs/1802.08219",[387],[416],{"type":24,"value":413},{"type":18,"tag":26,"props":418,"children":419},{},[420],{"type":24,"value":421},"Maurice Weiler, Mario Geiger, Max Welling, et al. 2018. 3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data. 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