Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometry
This work addresses the need for faster and more stable metrics in SPD neural networks, which is incremental as it builds on existing geometry insights to enhance practical applications.
The authors tackled the problem of designing efficient and stable Riemannian metrics for Symmetric Positive Definite (SPD) matrices by leveraging a product structure from Cholesky factors, resulting in two new metrics (PCM and BWCM) that offer closed-form operators, computational efficiency, and improved numerical stability.
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.