Riemannian Multiclass Logistics Regression for SPD Neural Networks

TL;DR — We design Riemannian multiclass logistics regression for SPD networks.

Abstract

Deep neural networks for learning Symmetric Positive Definite (SPD) matrices are gaining increasing attention in machine learning. Despite the significant progress, most existing SPD networks use traditional Euclidean classifiers on approximated spaces rather than intrinsic classifiers that accurately capture the geometry of SPD manifolds. Inspired by the success of Hyperbolic Neural Networks (HNNs), we propose Riemannian multiclass logistics regression (RMLR) for the classification layers in SPD networks. We focus on the metrics pulled back from the Euclidean space, such as Log-Euclidean Metric (LEM) and Log-Cholesky Metric (LCM), and introduce a unified framework for building Riemannian classifiers under these metrics. We first generalize the existing LEM and LCM by the concept of deformation and then design the specific SPD classifiers. Our framework encompasses the most popular LogEig classifier in existing SPD networks as a special case. The effectiveness of our method is demonstrated in three applications: radar recognition, human action recognition, and electroencephalography (EEG) classification.