Riemannian Networks over Full-Rank Correlation Matrices

TL;DR

This paper introduces Riemannian networks over the correlation manifold, extending neural network layers to this space and demonstrating their effectiveness through experiments.

Contribution

It systematically extends neural network layers to the correlation manifold using five correlation geometries, a largely unexplored area.

Findings

The proposed methods outperform existing SPD and Grassmannian networks.

New backpropagation techniques for correlation geometries are developed.

Experiments validate the effectiveness of Riemannian networks over correlation matrices.

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.