Fractional-Order Learning Dynamics on SPD Manifolds

In this study, we investigate the integration of fractional calculus and Riemannian geometry within the framework of machine learning. We propose a fractional-order learning model that operates on the Symmetric Positive Definite (SPD) manifold, a curved geometric space naturally suited for representing covariance-based data. Each data instance is modeled as an SPD matrix, ensuring that the learning dynamics respect the manifold’s intrinsic geometry. The UCI Human Activity Recognition (HAR) dataset was used to demonstrate the method, in which multichannel sensor recordings were transformed into SPD covariance representations. The learning algorithm extends classical gradient descent by incorporating a Caputo-type fractional derivative, introducing a controllable memory effect that enables updates to depend on both the current and historical gradients. Experiments performed for fractional orders ν=1.0, 1.2, and 1.5 show that increasing the fractional order yields smoother convergence trajectories, mitigates oscillations, and improves stability on the SPD manifold. This fractional–geometric coupling yields a regularized, memory-aware learning paradigm that enhances generalization capability, particularly for small or noisy datasets. The results suggest that fractional dynamics can act as an implicit temporal regularizer for Riemannian optimization, offering a principled approach to stabilize manifold-based learning algorithms. Overall, the study introduces a coherent framework that bridges fractional calculus, information geometry, and modern machine learning, and highlights the potential of fractional-order geometric learning as a unifying paradigm that links mathematical theory, computational modeling, and real-world sensor data analysis.