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 SPD neural networks. To remedy this limitation, we leverage the idea of pullback and propose adaptive Riemannian metrics for SPD manifolds. Moreover, we present comprehensive theories for our metrics. Experiments on three datasets demonstrate that equipped with the proposed metrics, SPD networks can exhibit superior performance.

翻译：对称正定矩阵因其内在能够编码数据底层结构相关性的能力，在机器学习领域受到广泛关注。为反映SPD流形的非欧几里得几何特性，研究者提出了多种有效的黎曼度量。然而，现有固定度量张量可能导致SPD矩阵学习（尤其是SPD神经网络）的性能次优。为弥补这一局限，我们利用拉回思想提出了SPD流形上的自适应黎曼度量，并给出了这些度量的完整理论框架。在三个数据集上的实验表明，采用所提度量的SPD网络能够展现出更优的性能。