基于同余结构的深度神经网络在正定矩阵上的表达能力
Expressivity of congruence-based architectures for DNNs on positive-definite matrices
AI总结：
研究同余层（输入矩阵左乘和右乘权重矩阵及其转置）在正定矩阵分类中的表达能力，发现半正交约束会限制网络表达能力，导致退化为单隐藏层等价结构，并分析了不同黎曼分类器与同余层特征图的兼容性。
摘要：
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，其中输入矩阵左乘和右乘一个（可能是矩形的）权重矩阵 $W$ 及其转置。这类层是著名的 SPDNet 的核心，也已被独立用于正定数据的降维。我们表明，通常对 $W$ 施加的（半）正交约束限制了这些层的表达能力：对于某些激活函数，生成的架构退化为单隐藏层等价结构。这种表达能力的缺失源于半正交 $W$ 的同余类层中谱多样性的损失，并且是庞加莱分离定理的直接结果。然后，我们考察了最终分类器的选择，比较了几种黎曼分类器，并讨论了它们与同余类层产生的特征图的兼容性。
英文摘要：
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.
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