We study algebraic neural networks (AlgNNs) with commutative algebras which unify diverse architectures such as Euclidean convolutional neural networks, graph neural networks, and group neural networks under the umbrella of algebraic signal processing. An AlgNN is a stacked layered information processing structure where each layer is conformed by an algebra, a vector space and a homomorphism between the algebra and the space of endomorphisms of the vector space. Signals are modeled as elements of the vector space and are processed by convolutional filters that are defined as the images of the elements of the algebra under the action of the homomorphism. We analyze stability of algebraic filters and AlgNNs to deformations of the homomorphism and derive conditions on filters that lead to Lipschitz stable operators. We conclude that stable algebraic filters have frequency responses – defined as eigenvalue domain representations – whose derivative is inversely proportional to the frequency – defined as eigenvalue magnitudes. It follows that for a given level of discriminability, AlgNNs are more stable than algebraic filters, thereby explaining their better empirical performance. This same phenomenon has been proven for Euclidean convolutional neural networks and graph neural networks. Our analysis shows that this is a deep algebraic property shared by a number of architectures.

中文翻译：

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代数神经网络：变形稳定性

我们研究具有交换代数的代数神经网络 (AlgNNs)，这些网络在代数信号处理的保护伞下统一了欧几里得卷积神经网络、图神经网络和群神经网络等多种架构。AlgNN 是一种堆叠的分层信息处理结构，其中每一层都由代数、向量空间以及代数与向量空间的自同态空间之间的同态构成。信号被建模为向量空间的元素，并由卷积滤波器处理，卷积滤波器定义为在同态作用下的代数元素的图像。我们分析代数滤波器和 AlgNN 对同态变形的稳定性，并推导出导致 Lipschitz 稳定算子的滤波器条件。我们得出结论，稳定代数滤波器具有频率响应——定义为特征值域表示——其导数与频率成反比——定义为特征值幅度。因此，对于给定的可辨别水平，AlgNNs 比代数滤波器更稳定，从而解释了它们更好的经验性能。欧几里得卷积神经网络和图神经网络已经证明了同样的现象。我们的分析表明，这是许多架构共享的深层代数属性。从而解释了他们更好的经验表现。欧几里得卷积神经网络和图神经网络已经证明了同样的现象。我们的分析表明，这是许多架构共享的深层代数属性。从而解释了他们更好的经验表现。欧几里得卷积神经网络和图神经网络已经证明了同样的现象。我们的分析表明，这是许多架构共享的深层代数属性。