Dimensionality Reduction of Dynamics on Lie Groups via Structure-Aware Canonical Correlation Analysis

Incorporating prior knowledge into a data-driven modeling problem can drastically improve performance, relia-bility, and generalization outside of the training sample. The stronger the structural properties, the more effective these improvements become. Manifolds are a powerful nonlinear generalization of Euclidean space for modeling finite dimensions. When additionally assuming that the manifold carries (Lie) group structure, this imposes a drastically stricter global constraint. The range of their applications is very wide and includes the important case of robotic tasks. We apply this idea to Canonical Correlation Analysis (CCA). In traditional CCA one constructs a hierarchical sequence of maximal correlations of up to two paired data sets in Euclidean spaces. We here generalize the CCA concept to respect the structure of Lie groups and demonstrate its efficacy through the substantial improvements it achieves in making structure-consistent pre-dictions about changes in the state of a robotic hand.