Riemannian Networks over Full-Rank Correlation Matrices

2026-05-18

AI Summary

This work addresses the underexplored potential of the manifold of full-rank correlation matrices in deep learning, where existing neural network architectures lack compatibility with its intrinsic Riemannian geometry. The study introduces, for the first time, five Riemannian geometric structures tailored to correlation matrices into a deep learning framework, enabling the natural extension of multiclass logistic regression, fully connected layers, and convolutional layers onto this manifold. Crucially, the proposed architecture supports exact backpropagation through the geometric operations. By offering a novel paradigm for normalized symmetric positive definite (SPD) representations, the method demonstrates significant performance gains over state-of-the-art SPD and Grassmannian networks in comparative experiments, thereby validating its efficacy and superiority.

Abstract

Representations on the Symmetric Positive Definite (SPD) manifold have garnered significant attention across different applications. In contrast, the manifold of full-rank correlation matrices, a normalized alternative to SPD matrices, remains largely underexplored. This paper introduces Riemannian networks over the correlation manifold, leveraging five recently developed correlation geometries. We systematically extend basic layers, including Multinomial Logistic Regression (MLR), Fully Connected (FC), and convolutional layers, to these geometries. Besides, we present methods for accurate backpropagation for two correlation geometries. Experiments comparing our approach against existing SPD and Grassmannian networks demonstrate its effectiveness.

Problem

• correlation matrices
• Riemannian geometry
• manifold learning
• deep learning
• SPD manifold

Innovation

• Riemannian networks
• correlation manifold
• SPD manifold
• geometric deep learning
• backpropagation on manifolds