Higher Order Gauge Equivariant CNNs on Riemannian Manifolds and Applications
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.
Code (0)
등록된 구현이 없습니다.
Methods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
Preserving gauge invariance in neural networks
In these proceedings we present lattice gauge equivariant convolutional neural networks (L-CNNs) which are able to process data from lattice gauge theory simulations while exactly preserving gauge symmetry. We review asp…
regressionLattice gauge symmetry in neural networks
We review a novel neural network architecture called lattice gauge equivariant convolutional neural networks (L-CNNs), which can be applied to generic machine learning problems in lattice gauge theory while exactly prese…
regressionGeometrical aspects of lattice gauge equivariant convolutional neural networks
Lattice gauge equivariant convolutional neural networks (L-CNNs) are a framework for convolutional neural networks that can be applied to non-Abelian lattice gauge theories without violating gauge symmetry. We demonstrat…
Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds
Motivated by the vast success of deep convolutional networks, there is a great interest in generalizing convolutions to non-Euclidean manifolds. A major complication in comparison to flat spaces is that it is unclear in …
Applications of Lattice Gauge Equivariant Neural Networks
The introduction of relevant physical information into neural network architectures has become a widely used and successful strategy for improving their performance. In lattice gauge theories, such information can be ide…