Adaptive Riemannian Metrics on SPD Manifolds

Symmetric Positive Definite (SPD) matrices have received wide attention in machine learning due to their intrinsic capacity of encoding underlying structural correlation in data. To reflect the non-Euclidean geometry of SPD manifolds, many successful Riemannian metrics have been proposed. However, existing fixed metric tensors might lead to sub-optimal performance for SPD matrices learning, especially for deep SPD neural networks. To remedy this limitation, we leverage the commonly encountered pullback techniques, and propose adaptive Riemannian metrics for SPD manifolds. Moreover, we present comprehensive theories to support our metrics. The experimental and theoretical analysis demonstrates that the merit of the proposed metrics, in optimising SPD network framework with promising performance.