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.

翻译：本文研究用于分类对称正定矩阵的神经架构，重点关注同余类层——即输入矩阵左乘和右乘（可能为矩形的）权重矩阵$W$及其转置。此类层是著名的SPDNet的核心组件，也已被独立用于正定数据的降维。我们证明，通常施加于$W$的（半）正交约束限制了这些层的表达能力：对于某些激活函数，所得架构会坍缩为等价于单隐藏层的结构。这种表达能力的缺失源于半正交$W$的同余类层丧失了谱多样性，且是庞加莱分离定理的直接推论。进而，我们考察最终分类器的选择，比较了若干黎曼分类器，并讨论了它们与同余类层产生的特征图的兼容性。