Riemannian Local Mechanism for SPD Neural Networks

TL;DR

This paper introduces a novel Riemannian local mechanism for SPD neural networks that explicitly captures local geometric information on the SPD manifold, improving data representation for visual tasks.

Contribution

It proposes a new multi-scale submanifold block that preserves local geometry in SPD networks, inspired by category theory analysis of Euclidean local mechanisms.

Findings

Enhanced local geometric information preservation in SPD networks.

Improved performance on multiple visual tasks.

Validated effectiveness through experiments.

Abstract

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.