Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry

TL;DR

This paper investigates why matrix function normalizations like logarithm and power improve covariance pooling in deep neural networks, revealing their connection to Riemannian geometry and classifiers on the SPD manifold.

Contribution

It provides a unified Riemannian geometric explanation for matrix function normalizations in covariance pooling, bridging tangent and Riemannian classifiers.

Findings

Matrix functions implicitly respect Riemannian classifiers.

Theoretical analysis aligns with empirical results.

Normalizations improve classification by respecting manifold geometry.

Abstract

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. The code is available at https://github.com/GitZH-Chen/RiemGCP.git.