Sparse Bayesian learning for dynamical modelling on product manifolds

Abstract

In imitation learning, Bayesian approaches are widely applied for encoding robotic skills. However, most existing works focus on tasks represented in Euclidean spaces, which cannot properly characterize non-Euclidean behaviours such as robot orientation. In this paper, we propose an intrinsic Bayesian scheme for learning dynamical models on product manifolds, enabling effective learning of pose-related tasks. First, an intrinsic weighted-metric is presented for statistical analysis on product manifolds. Its validity is rigorously proven by satisfying metric axioms. Then, to decouple the constrained multi-output system without increasing computational complexity, a manifold dynamical model of the sub-system is proposed using parallel transport across local charts, ensuring geometric consistency. After that, the manifold Gaussian process is developed by incorporating the intrinsic weighted metric, significantly improving regression accuracy. But the computational complexity of this approach is constrained by the size of the covariance matrix, particularly for large datasets. To further enhance the calculation efficiency, manifold sparse Bayesian learning is proposed by considering sparse priors. Finally, simulations and experimental studies show the effectiveness and accuracy of the proposed Bayesian scheme on product manifolds.