论文标题
Wigner-Eckart定理用于群体卷积内核
A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels
论文作者
论文摘要
小组模棱两可的卷积网络（GCNNS）具有其他对称性先验的endow经典卷积网络，这可能会导致性能大大提高。 GCNNS的理论描述的最新进展表明，这种模型通常可以理解为与G-Steerable内核进行卷积，即满足均衡性约束本身的内核。尽管已得出了G-Stererability的约束，但它仅针对特定用例就解决了 - g-steerable内核空间的一般表征仍然缺失。这项工作为g成为任何紧凑型组的实际相关案例提供了这种特征。另一方面，我们的调查是由限制下的限制与量子力学的球形张量运算符之间的惊人类比进行的。通过概括著名的球形张量操作员的Wigner-Eckart定理，我们证明可通话的内核空间是完全理解和参数为1）概括的矩阵元素的，2）Clebsch-Gordan系数，以及3）均匀空间上的谐波基础函数。
Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with additional symmetry priors, which can lead to a considerably improved performance. Recent advances in the theoretical description of GCNNs revealed that such models can generally be understood as performing convolutions with G-steerable kernels, that is, kernels that satisfy an equivariance constraint themselves. While the G-steerability constraint has been derived, it has to date only been solved for specific use cases - a general characterization of G-steerable kernel spaces is still missing. This work provides such a characterization for the practically relevant case of G being any compact group. Our investigation is motivated by a striking analogy between the constraints underlying steerable kernels on the one hand and spherical tensor operators from quantum mechanics on the other hand. By generalizing the famous Wigner-Eckart theorem for spherical tensor operators, we prove that steerable kernel spaces are fully understood and parameterized in terms of 1) generalized reduced matrix elements, 2) Clebsch-Gordan coefficients, and 3) harmonic basis functions on homogeneous spaces.
