The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks

• URL: http://arxiv.org/abs/2407.07655v1
• Date: Wed, 10 Jul 2024 13:35:04 GMT
• Title: The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks
• Authors: Simon Mataigne, Johan Mathe, Sophia Sanborn, Christopher Hillar, Nina Miolane,
• Abstract summary: We show that the $G$-Bispectrum can be reduced into a textitselective $G$-Bispectrum. We demonstrate how its integration in neural networks enhances accuracy and robustness compared to traditional approaches.
• Score: 3.8311785959108637
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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