  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.
