An important problem in signal processing and deep learning is to achieve \textit{invariance} to nuisance factors not relevant for the task. Since many of these factors are describable as the action of a group $G$ (e.g. rotations, translations, scalings), we want methods to be $G$-invariant. The $G$-Bispectrum extracts every characteristic of a given signal up to group action: for example, the shape of an object in an image, but not its orientation. Consequently, the $G$-Bispectrum has been incorporated into deep neural network architectures as a computational primitive for $G$-invariance\textemdash akin to a pooling mechanism, but with greater selectivity and robustness. However, the computational cost of the $G$-Bispectrum ($\mathcal{O}(|G|^2)$, with $|G|$ the size of the group) has limited its widespread adoption. Here, we show that the $G$-Bispectrum computation contains redundancies that can be reduced into a \textit{selective $G$-Bispectrum} with $\mathcal{O}(|G|)$ complexity. We prove desirable mathematical properties of the selective $G$-Bispectrum and demonstrate how its integration in neural networks enhances accuracy and robustness compared to traditional approaches, while enjoying considerable speeds-up compared to the full $G$-Bispectrum.

翻译：信号处理和深度学习中的一个重要问题是实现对任务无关的干扰因素的\textit{不变性}。由于许多此类因素可描述为某个群 $G$（例如旋转、平移、缩放）的作用，我们希望方法具有 $G$ 不变性。$G$-双谱提取给定信号在群作用下的所有特征：例如，图像中物体的形状，而非其方向。因此，$G$-双谱已被整合到深度神经网络架构中，作为实现 $G$ 不变性的计算原语——类似于池化机制，但具有更高的选择性和鲁棒性。然而，$G$-双谱的计算成本（$\mathcal{O}(|G|^2)$，其中 $|G|$ 为群的大小）限制了其广泛应用。本文中，我们证明了 $G$-双谱计算包含冗余，可将其简化为具有 $\mathcal{O}(|G|)$ 复杂度的\textit{选择性 $G$-双谱}。我们证明了选择性 $G$-双谱具备理想的数学性质，并展示了其在神经网络中的集成如何相较于传统方法提高准确性和鲁棒性，同时相比完整的 $G$-双谱实现了显著的速度提升。