The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915006779490304 |
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| author | Mataigne, Simon Mathe, Johan Sanborn, Sophia Hillar, Christopher Miolane, Nina |
| author_facet | Mataigne, Simon Mathe, Johan Sanborn, Sophia Hillar, Christopher Miolane, Nina |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07655 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks Mataigne, Simon Mathe, Johan Sanborn, Sophia Hillar, Christopher Miolane, Nina Machine Learning 68T01, 68T07, 68R01, 20K01 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. |
| title | The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks |
| topic | Machine Learning 68T01, 68T07, 68R01, 20K01 |
| url | https://arxiv.org/abs/2407.07655 |