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.

翻译：对称正定矩阵因其内在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，已有多种成功的黎曼度量被提出。然而，现有度量张量大多固定不变，可能导致SPD矩阵学习（特别是深度SPD神经网络）的性能欠佳。为克服此局限，我们利用常见的拉回技术，提出了自适应对数欧几里得度量，该方法是广泛应用的对数欧几里得度量的扩展。相较于既有黎曼度量，我们的度量包含可学习参数，能以微小的额外计算量更好地适应黎曼神经网络的复杂动态特性。我们还提供了完整的理论分析以支撑所提出的度量方法，包括代数性质与黎曼性质。实验与理论结果均证明了所提度量在提升SPD神经网络性能方面的优势。该度量的有效性在一系列最新开发的黎曼构建模块中得到进一步验证，包括黎曼批量归一化、黎曼残差块和黎曼分类器。