{"ID":2923531,"CreatedAt":"2026-06-02T04:05:25.881865328Z","UpdatedAt":"2026-06-04T13:12:39.622923895Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.02490","arxiv_id":"2606.02490","title":"Expressivity of congruence-based architectures for DNNs on positive-definite matrices","abstract":"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.","short_abstract":"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 al...","url_abs":"https://arxiv.org/abs/2606.02490","url_pdf":"https://arxiv.org/pdf/2606.02490v1","authors":"[\"Antonin Oswald\",\"Estelle Massart\"]","published":"2026-06-01T17:01:08Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[]","has_code":false}