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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2606.02490 |
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| _version_ | 1866910282360553472 |
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| author | Oswald, Antonin Massart, Estelle |
| author_facet | Oswald, Antonin Massart, Estelle |
| contents | This work studies neural architectures for classifying symmetric positive-definite matrices, focusing on congruence-like layers, in which the input matrix is multiplied on the left and right by a (possibly rectangular) weight matrix $W$ and its transpose. Such layers lie at the core of the celebrated SPDNet and have also been employed independently for dimensionality reduction on positive-definite data. We show that the (semi)-orthogonality constraint commonly imposed on $W$ limits the expressivity of these layers: for certain activation functions, the resulting architecture collapses to a one-hidden-layer equivalent. This lack of expressivity follows from a loss of spectral diversity in congruence-like layers for semi-orthogonal $W$ and is a direct consequence of Poincaré's separation theorem. We then examine the choice of the final classifier, comparing several Riemannian classifiers and discussing their compatibility with the feature maps produced by congruence-like layers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02490 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Expressivity of congruence-based architectures for DNNs on positive-definite matrices Oswald, Antonin Massart, Estelle Machine Learning This work studies neural architectures for classifying symmetric positive-definite matrices, focusing on congruence-like layers, in which the input matrix is multiplied on the left and right by a (possibly rectangular) weight matrix $W$ and its transpose. Such layers lie at the core of the celebrated SPDNet and have also been employed independently for dimensionality reduction on positive-definite data. We show that the (semi)-orthogonality constraint commonly imposed on $W$ limits the expressivity of these layers: for certain activation functions, the resulting architecture collapses to a one-hidden-layer equivalent. This lack of expressivity follows from a loss of spectral diversity in congruence-like layers for semi-orthogonal $W$ and is a direct consequence of Poincaré's separation theorem. We then examine the choice of the final classifier, comparing several Riemannian classifiers and discussing their compatibility with the feature maps produced by congruence-like layers. |
| title | Expressivity of congruence-based architectures for DNNs on positive-definite matrices |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2606.02490 |