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

TL;DR

This paper introduces two novel Riemannian metrics for SPD matrices, derived from Cholesky factorization, offering improved computational efficiency and numerical stability for SPD neural network applications.

Contribution

We propose the Power–Cholesky and Bures–Wasserstein–Cholesky metrics, leveraging Cholesky product geometry for faster, more stable SPD matrix computations in neural networks.

Findings

The new metrics outperform existing ones in stability and efficiency.

They enable effective SPD neural network components like classifiers and residual blocks.

Experiments confirm robustness and practical advantages.

Abstract

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.