  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.
