Group equivariant convolutional neural networks (G-CNNs) are generalizations
of convolutional neural networks (CNNs) which excel in a wide range of
technical applications by explicitly encoding symmetries, such as rotations and
permutations, in their architectures. Although the success of G-CNNs is driven
by their \emph{explicit} symmetry bias, a recent line of work has proposed that
the \emph{implicit} bias of training algorithms on particular architectures is
key to understanding generalization for overparameterized neural nets. In this
context, we show that $L$-layer full-width linear G-CNNs trained via gradient
descent for binary classification converge to solutions with low-rank Fourier
matrix coefficients, regularized by the $2/L$-Schatten matrix norm. Our work
strictly generalizes previous analysis on the implicit bias of linear CNNs to
linear G-CNNs over all finite groups, including the challenging setting of
non-commutative groups (such as permutations), as well as band-limited G-CNNs
over infinite groups. We validate our theorems via experiments on a variety of
groups, and empirically explore more realistic nonlinear networks, which
locally capture similar regularization patterns. Finally, we provide intuitive
interpretations of our Fourier space implicit regularization results in real
space via uncertainty principles.

本文研究群等变卷积神经网络，在训练时隐式地通过具体的架构对模型进行正则化处理来实现其显式的对称性偏置，从而实现模型的推广。同时，文中还提出了傅里叶空间隐式正则化模型的解释，并通过实验证明了该模型的有效性。