Symmetric positive definite (SPD) matrix has been demonstrated to be an effective feature descriptor in many scientific areas, as it can encode spatiotemporal statistics of the data adequately on a curved Riemannian manifold, i.e., SPD manifold. Although there are many different ways to design network architectures for SPD matrix nonlinear learning, very few solutions explicitly mine the geometrical dependencies of features at different layers. Motivated by the great success of self-attention mechanism in capturing long-range relationships, an SPD manifold self-attention mechanism (SMSA) is proposed in this paper using some manifold-valued geometric operations, mainly the Riemannian metric, Riemannian mean, and Riemannian optimization. Then, an SMSA-based geometric learning module (SMSA-GLM) is designed for the sake of improving the discrimination of the generated deep structured representations. Extensive experimental results achieved on three benchmarking datasets show that our modification against the baseline network further alleviates the information degradation problem and leads to improved accuracy.

翻译：对称正定（SPD）矩阵已被证明在许多科学领域中是一种有效的特征描述符，因为它能够充分地在弯曲的黎曼流形（即SPD流形）上编码数据的时空统计信息。尽管存在多种设计用于SPD矩阵非线性学习的网络架构的方法，但很少有解决方案能够显式地挖掘不同层特征的几何依赖性。受自注意力机制在捕获长距离关系方面巨大成功的启发，本文利用多种流形值几何运算（主要包括黎曼度量、黎曼均值和黎曼优化）提出了一种SPD流形自注意力机制（SMSA）。随后，设计了一种基于SMSA的几何学习模块（SMSA-GLM），以提高所生成深度结构化表示的判别能力。在三个基准数据集上取得的广泛实验结果表明，我们对基线网络所做的修改进一步缓解了信息退化问题，并提升了准确率。