黎曼流形上的高阶规范等变CNN及其应用
Higher Order Gauge Equivariant CNNs on Riemannian Manifolds and Applications
AI总结：
本文提出高阶规范等变卷积及其实现GEVNet，在黎曼流形上建模感受野内非线性交互并保持全局等距等变性，结合CKN在dMRI神经影像分类中超越单一组件性能。
摘要：
随着深度网络文献中群等变卷积的出现，具有$\mathsf{SO}(3)$等变层的球面CNN已被开发出来，用于处理球面$S^2$上信号采样数据。通过显式要求关于$\mathsf{SO}(2)$的规范等变性，可以在$S^2$上隐式获得$\mathsf{SO}(3)$等变卷积，并带来显著的效率提升。本文在此基础上引入规范等变卷积的高阶推广，其实现被称为规范等变Volterra网络（GEVNet）。这使我们能够在给定感受野内建模空间扩展的非线性交互，同时保持对全局等距变换的等变性。我们证明了关于高阶规范等变卷积的等变性和构造的理论结果。然后，我们通过实验证明了模型的参数效率，首先在计算机视觉基准数据（如球面MNIST）上，然后在神经影像数据上与卷积核网络（CKN）结合使用。在神经影像数据实验中，所得到的两部分架构（CKN + GEVNet）被用于从扩散磁共振图像（dMRI）中自动区分路易体病（DLB）、阿尔茨海默病（AD）和帕金森病（PD）患者。GEVNet提取每个体素内的微结构特征，而CKN提取跨体素的宏观结构特征。这种复合架构独特地能够利用dMRI数据中包含的体素内和体素间信息，从而在分类结果上优于任一单独组件所获得的性能。
英文摘要：
With the advent of group equivariant convolutions in deep networks literature, spherical CNNs with $\mathsf{SO}(3)$-equivariant layers have been developed to cope with data that are samples of signals on the sphere $S^2$. One can implicitly obtain $\mathsf{SO}(3)$-equivariant convolutions on $S^2$ with significant efficiency gains by explicitly requiring gauge equivariance w.r.t. $\mathsf{SO}(2)$. In this paper, we build on this fact by introducing a higher order generalization of the gauge equivariant convolution, whose implementation is dubbed a gauge equivariant Volterra network (GEVNet). This allows us to model spatially extended nonlinear interactions within a given receptive field while still maintaining equivariance to global isometries. We prove theoretical results regarding the equivariance and construction of higher order gauge equivariant convolutions. Then, we empirically demonstrate the parameter efficiency of our model, first on computer vision benchmark data (e.g. spherical MNIST), and then in combination with a convolutional kernel network (CKN) on neuroimaging data. In the neuroimaging data experiments, the resulting two-part architecture (CKN + GEVNet) is used to automatically discriminate between patients with Lewy Body Disease (DLB), Alzheimer's Disease (AD) and Parkinson's Disease (PD) from diffusion magnetic resonance images (dMRI). The GEVNet extracts micro-architectural features within each voxel, while the CKN extracts macro-architectural features across voxels. This compound architecture is uniquely poised to exploit the intra- and inter-voxel information contained in the dMRI data, leading to improved performance over the classification results obtained from either of the individual components.
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