The Symmetric Positive Definite (SPD) matrices have received wide attention for data representation in many scientific areas. Although there are many different attempts to develop effective deep architectures for data processing on the Riemannian manifold of SPD matrices, very few solutions explicitly mine the local geometrical information in deep SPD feature representations. Given the great success of local mechanisms in Euclidean methods, we argue that it is of utmost importance to ensure the preservation of local geometric information in the SPD networks. We first analyse the convolution operator commonly used for capturing local information in Euclidean deep networks from the perspective of a higher level of abstraction afforded by category theory. Based on this analysis, we define the local information in the SPD manifold and design a multi-scale submanifold block for mining local geometry. Experiments involving multiple visual tasks validate the effectiveness of our approach. The supplement and source code can be found in https://github.com/GitZH-Chen/MSNet.git.

翻译：对称正定（SPD）矩阵因其在众多科学领域中用于数据表示而受到广泛关注。尽管已有多种不同尝试致力于开发在SPD矩阵黎曼流形上进行数据处理的深度架构，但很少有解决方案能显式挖掘深层SPD特征表示中的局部几何信息。鉴于局部机制在欧几里得方法中取得的巨大成功，我们认为确保SPD网络中局部几何信息的保留至关重要。我们首先从范畴论所提供的更高抽象层次出发，分析了欧几里得深度网络中常用于捕获局部信息的卷积算子。基于此分析，我们定义了SPD流形中的局部信息，并设计了一个多尺度子流形模块用于挖掘局部几何结构。涉及多个视觉任务的实验验证了所提方法的有效性。补充材料和源代码可在https://github.com/GitZH-Chen/MSNet.git获取。