从黎曼几何视角理解协方差池化中的矩阵函数归一化
Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry
AI总结：
本文从黎曼几何视角统一解释了全局协方差池化中矩阵对数与矩阵幂归一化的作用机制，指出其有效性源于隐含的黎曼分类器，并通过细粒度与大规模视觉分类实验验证了该结论。
摘要：
全局协方差池化（GCP）通过利用高层表示的二阶统计量，已被证明能够提升深度神经网络（DNN）的性能。GCP通常通过对协方差矩阵应用矩阵函数归一化（如矩阵对数或矩阵幂），然后接一个欧氏分类器来进行分类。然而，协方差矩阵本质上位于一个黎曼流形上，即对称正定（SPD）流形。现有文献未能对“为什么在矩阵幂归一化之后，欧氏分类器可以直接应用于黎曼特征”这一问题给出令人满意的解释。为弥补这一空白，本文从黎曼几何视角对矩阵对数和矩阵幂提供了全面且统一的理解。GCP中矩阵函数的底层机制从两个角度进行解释：一个基于切空间分类器（切空间上的欧氏分类器），另一个基于黎曼分类器。通过在细粒度和大规模视觉分类数据集上进行理论分析和大量实验的经验验证，我们得出结论：矩阵函数的工作机制应归因于它们隐含遵循的黎曼分类器。代码可在 https://github.com/GitZH-Chen/RiemGCP.git 获取。
英文摘要：
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.
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