Riemannian Networks over Full-Rank Correlation Matrices

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.

Many types of data can be described by how different measurements relate to one another. These relationships are often stored in correlation matrices. This paper develops neural network layers that learn from correlation matrices while respecting their geometry. Experiments show that it can be more accurate and efficient than related covariance-based or matrix-based methods. These layers provide useful building blocks for future models that learn from correlation patterns.