Generalization Bounds for Parameterized Group Equivariant Convolutional Neural Networks

Equivariant convolutional neural networks have shown strong empirical performance by incorporating symmetry into their architectures, yet their generalization behavior remains theoretically underexplored, particularly for parameterized group equivariant CNNs, where convolutional filters are represented as linear combinations of basis functions. In this paper, we derive explicit generalization error bounds that characterize the effects of network depth, channel width, kernel size, and group size. Our bounds further capture the effects of filter parameterization, including the number of basis functions, coefficient constraints, and convolution-operator spectral norms, providing quantitative insights into their contributions to model complexity and generalization error. In particular, our analysis shows that different basis parameterizations can lead to different generalization guarantees, even for the same network architecture, and that reducing the effective basis dimension can improve the resulting generalization bound. These results provide a theoretical basis for understanding how filter parameterization affects the generalization of parameterized equivariant CNNs.