Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometry

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.

💡 Research Summary

This paper addresses the longstanding challenge of designing efficient and numerically stable Riemannian metrics for Symmetric Positive Definite (SPD) matrices, which are central to many modern machine‑learning applications such as medical imaging, EEG analysis, and computer vision. While several metrics—Affine‑Invariant (AIM), Log‑Euclidean (LEM), Power‑Euclidean (PEM), Log‑Cholesky (LCM), Bures‑Wasserstein (BWM), and Generalized Bures‑Wasserstein (GBWM)—have been proposed, each suffers from either computational overhead, lack of closed‑form operators, or numerical instability (especially due to matrix logarithms).

The authors revisit the geometry of the Cholesky factors of SPD matrices. They observe that a Cholesky matrix (L) can be uniquely decomposed into a strictly lower‑triangular part (\lfloor L\rfloor) (an Euclidean space) and a diagonal part (\operatorname{diag}(L)) (the product of (n) copies of the one‑dimensional positive real line). This yields a natural product manifold structure: \