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

• URL: http://arxiv.org/abs/2407.10484v2
• Date: Sat, 20 Jul 2024 08:11:10 GMT
• Title: Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry
• Authors: Ziheng Chen, Yue Song, Xiao-Jun Wu, Gaowen Liu, Nicu Sebe,
• Abstract summary: 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.
• Score: 63.694184882697435
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

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