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.

该论文提出了一种基于 Riemann 度量、Riemann 均值和 Riemann 优化的 SPD 流形自注意机制，用于改善所生成的深度结构表示的区分度，实验结果表明，该方法进一步减轻了信息退化问题并提高了准确性。

Riemannian 自我注意力机制用于 SPD 网络

Riemannian Self-Attention Mechanism for SPD Networks

Representing images and videos with Symmetric Positive Definite (SPD)
matrices and considering the Riemannian geometry of the resulting space has
proven beneficial for many recognition tasks. Unfortunately, computation on the
Riemannian manifold of SPD matrices --especially of high-dimensional ones--
comes at a high cost that limits the applicability of existing techniques. In
this paper we introduce an approach that lets us handle high-dimensional SPD
matrices by constructing a lower-dimensional, more discriminative SPD manifold.
To this end, we model the mapping from the high-dimensional SPD manifold to the
low-dimensional one with an orthonormal projection. In particular, we search
for a projection that yields a low-dimensional manifold with maximum
discriminative power encoded via an affinity-weighted similarity measure based
on metrics on the manifold. Learning can then be expressed as an optimization
problem on a Grassmann manifold. Our evaluation on several classification tasks
shows that our approach leads to a significant accuracy gain over
state-of-the-art methods.

使用正定对称 (SPD) 矩阵表示图像和视频，并考虑到所得空间的里曼尼几何，已被证明在许多识别任务中有益。本文引入了一种方法来构建一个更具判别力的低维 SPD 流形以处理高维 SPD 矩阵，并将学习表述为 Grassmann 流形上的优化问题。实验表明，与现有技术相比，我们的方法可使分类准确性显著提高。