Incorporating prior knowledge into a data-driven modeling problem can drastically improve performance, reliability, 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. Structural impositions in constrained systems increase when applying group structure, converting them into Lie manifolds. The range of their applications is very wide and includes the important case of robotic tasks. Canonical Correlation Analysis (CCA) can construct a hierarchical sequence of maximal correlations of up to two paired data sets in these Euclidean spaces. We present a method to generalize this concept to Lie Manifolds and demonstrate its efficacy through the substantial improvements it achieves in making structure-consistent predictions about changes in the state of a robotic hand.

翻译：将先验知识融入数据驱动建模问题中，可显著提升训练样本外的性能、可靠性和泛化能力。结构属性越强，这些改进的效果越显著。流形作为欧几里得空间的一种强大非线性推广，适用于有限维建模。当引入群结构时，约束系统中的结构性会进一步增强，从而将其转化为李流形。此类流形的应用范围极为广泛，其中包含机器人任务这一重要场景。典型相关分析（CCA）能在欧几里得空间中构建最多两组配对数据层级序列的最大相关性。我们提出了一种将该概念推广至李流形的方法，并通过其在机器人手部状态变化的结构一致性预测中取得的显著改进，验证了该方法的有效性。