在SPD流形上通过Cholesky积几何实现快速且稳定的Riemannian度量
Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometry
AI总结：
本文提出两种快速稳定的SPD度量，通过Cholesky分解设计，提升SPD神经网络的计算效率和数值稳定性。
摘要：
最近在对称正定（SPD）矩阵学习中的进展表明，Riemannian度量是有效SPD神经网络的关键。受此启发，我们重新审视Cholesky因子的几何结构，并揭示了一种简单的积结构，使度量设计更加方便。基于这一见解，我们提出了两种快速且稳定的SPD度量，即幂Cholesky度量（PCM）和Bures-Wasserstein-Cholesky度量（BWCM），通过Cholesky分解推导而来。与现有SPD度量相比，所提度量提供了闭式运算符、计算效率和改进的数值稳定性。我们进一步将这些度量应用于构造Riemannian多项式逻辑回归（MLR）分类器和SPD神经网络的残差块。在SPD深度学习、数值稳定性分析和张量插值实验中，证明了所提度量的有效性、效率和鲁棒性。代码可在https://github.com/GitZH-Chen/PCM_BWCM获取。
英文摘要：
Recent advances in Symmetric Positive Definite (SPD) matrix learning show that Riemannian metrics are fundamental to effective SPD neural networks. Motivated by this, we revisit the geometry of the Cholesky factors and uncover a simple product structure that enables convenient metric design. Building on this insight, we propose two fast and stable SPD metrics, Power--Cholesky Metric (PCM) and Bures--Wasserstein--Cholesky Metric (BWCM), derived via Cholesky decomposition. Compared with existing SPD metrics, the proposed metrics provide closed-form operators, computational efficiency, and improved numerical stability. We further apply our metrics to construct Riemannian Multinomial Logistic Regression (MLR) classifiers and residual blocks for SPD neural networks. Experiments on SPD deep learning, numerical stability analyses, and tensor interpolation demonstrate the effectiveness, efficiency, and robustness of our metrics. The code is available at https://github.com/GitZH-Chen/PCM_BWCM.
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