Deep neural networks for learning Symmetric Positive Definite (SPD) matrices are gaining increasing attention in machine learning. Despite the significant progress, most existing SPD networks use traditional Euclidean classifiers on an approximated space rather than intrinsic classifiers that accurately capture the geometry of SPD manifolds. Inspired by Hyperbolic Neural Networks (HNNs), we propose Riemannian Multinomial Logistics Regression (RMLR) for the classification layers in SPD networks. We introduce a unified framework for building Riemannian classifiers under the metrics pulled back from the Euclidean space, and showcase our framework under the parameterized Log-Euclidean Metric (LEM) and Log-Cholesky Metric (LCM). Besides, our framework offers a novel intrinsic explanation for the most popular LogEig classifier in existing SPD networks. The effectiveness of our method is demonstrated in three applications: radar recognition, human action recognition, and electroencephalography (EEG) classification. The code is available at https://github.com/GitZH-Chen/SPDMLR.git.

翻译：学习对称正定（SPD）矩阵的深度神经网络在机器学习领域正日益受到关注。尽管取得了显著进展，但大多数现有SPD网络在近似空间上使用传统的欧几里得分类器，而非能够准确捕捉SPD流形几何结构的内在分类器。受双曲神经网络（HNNs）的启发，我们提出用于SPD网络分类层的黎曼多项逻辑回归（RMLR）。我们引入了一个统一框架，用于在从欧几里得空间拉回的度量下构建黎曼分类器，并在参数化对数欧几里得度量（LEM）和对数乔列斯基度量（LCM）下展示了该框架。此外，我们的框架为现有SPD网络中最常用的LogEig分类器提供了一种新颖的内在解释。该方法的有效性在三个应用案例中得到验证：雷达识别、人体动作识别和脑电图（EEG）分类。代码可在 https://github.com/GitZH-Chen/SPDMLR.git 获取。