Symmetric Positive Definite (SPD) matrices have received wide attention in machine learning due to their intrinsic capacity to encode underlying structural correlation in data. Many successful Riemannian metrics have been proposed to reflect the non-Euclidean geometry of SPD manifolds. However, most existing metric tensors are fixed, which might lead to sub-optimal performance for SPD matrix learning, especially for deep SPD neural networks. To remedy this limitation, we leverage the commonly encountered pullback techniques and propose Adaptive Log-Euclidean Metrics (ALEMs), which extend the widely used Log-Euclidean Metric (LEM). Compared with the previous Riemannian metrics, our metrics contain learnable parameters, which can better adapt to the complex dynamics of Riemannian neural networks with minor extra computations. We also present a complete theoretical analysis to support our ALEMs, including algebraic and Riemannian properties. The experimental and theoretical results demonstrate the merit of the proposed metrics in improving the performance of SPD neural networks. The efficacy of our metrics is further showcased on a set of recently developed Riemannian building blocks, including Riemannian batch normalization, Riemannian Residual blocks, and Riemannian classifiers.

Accepted by TIP 2024