  Global Covariance Pooling (GCP) has been demonstrated to improve the
performance of Deep Neural Networks (DNNs) by exploiting second-order
statistics of high-level representations. GCP typically performs classification
of the covariance matrices by applying matrix function normalization, such as
matrix logarithm or power, followed by a Euclidean classifier. However,
covariance matrices inherently lie in a Riemannian manifold, known as the
Symmetric Positive Definite (SPD) manifold. The current literature does not
provide a satisfactory explanation of why Euclidean classifiers can be applied
directly to Riemannian features after the normalization of the matrix power. To
mitigate this gap, this paper provides a comprehensive and unified
understanding of the matrix logarithm and power from a Riemannian geometry
perspective. The underlying mechanism of matrix functions in GCP is interpreted
from two perspectives: one based on tangent classifiers (Euclidean classifiers
on the tangent space) and the other based on Riemannian classifiers. Via
theoretical analysis and empirical validation through extensive experiments on
fine-grained and large-scale visual classification datasets, we conclude that
the working mechanism of the matrix functions should be attributed to the
Riemannian classifiers they implicitly respect.
