We describe generalizations of the universal approximation theorem for neural networks to maps invariant or equivariant with respect to linear representations of groups. Our goal is to establish network-like computational models that are both invariant/equivariant and provably complete in the sense of their ability to approximate any continuous invariant/equivariant map. Our contribution is three-fold. First, in the general case of compact groups we propose a construction of a complete invariant/equivariant network using an intermediate polynomial layer. We invoke classical theorems of Hilbert and Weyl to justify and simplify this construction; in particular, we describe an explicit complete ansatz for approximation of permutation-invariant maps. Second, we consider groups of translations and prove several versions of the universal approximation theorem for convolutional networks in the limit of continuous signals on euclidean spaces. Finally, we consider 2D signal transformations equivariant with respect to the group SE(2) of rigid euclidean motions. In this case we introduce the “charge–conserving convnet”—a convnet-like computational model based on the decomposition of the feature space into isotypic representations of SO(2). We prove this model to be a universal approximator for continuous SE(2)—equivariant signal transformations.

我们描述了神经网络的通用逼近定理的一般化，以相对于组的线性表示来映射不变或等变。我们的目标是建立类似网络的计算模型，该模型既不变/均等，又在逼近任何连续不变/等价图的能力方面可证明是完备的。我们的贡献是三倍。首先，在紧凑群的一般情况下，我们建议使用中间多项式层构造一个完整的不变/等变网络。我们引用希尔伯特和韦尔的经典定理来证明和简化这种构造。特别是，我们为置换不变映射的近似描述了一个显式的完整ansatz。第二，我们考虑了平移组，并证明了在欧几里德空间上连续信号的极限下，卷积网络的通用逼近定理的几种版本。最后，我们考虑相对于刚性欧几里得运动的SE（2）组等价的2D信号变换。在这种情况下，我们引入“电荷保持卷积” —一种类似于卷积的计算模型，该模型基于将特征空间分解为SO（2）的同型表示形式。我们证明该模型是连续SE（2）-等变信号转换的通用逼近器。在这种情况下，我们引入“电荷保持卷积” —一种类似于卷积的计算模型，该模型基于将特征空间分解为SO（2）的同型表示形式。我们证明该模型是连续SE（2）-等变信号转换的通用逼近器。在这种情况下，我们引入“电荷保持卷积” —一种类似于卷积的计算模型，该模型基于将特征空间分解为SO（2）的同型表示形式。我们证明该模型是连续SE（2）-等变信号转换的通用逼近器。