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：信号处理与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群 $G$ 的作用（例如旋转、平移、缩放），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，而非其朝向。因此，$G$-双谱已被作为实现 $G$-不变性的计算基元集成到深度神经网络架构中——类似于池化机制，但具有更强的选择性和鲁棒性。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。本文证明 $G$-双谱计算存在冗余，可简化为具有 $\mathcal{O}(|G|)$ 复杂度的\textit{选择性 $G$-双谱}。我们证明了选择性 $G$-双谱的理想数学特性，并通过实验表明其在神经网络中的集成相较于传统方法能提升准确性与鲁棒性，同时相比完整 $G$-双谱获得显著加速。