Data-driven approximations of the infinite-dimensional Koopman operator rely on finite-dimensional projections, where the predictive accuracy of the resulting models hinges heavily on the invariance of the chosen subspace. Subspace pruning systematically discards geometrically misaligned directions to enhance this invariance proximity, which formally corresponds to the largest principal angle between the subspace and its image under the operator. Yet, existing techniques are largely restricted to Euclidean settings. To bridge this gap, this paper presents an approach for computing principal angles and vectors to enable Koopman subspace pruning within a Reproducing Kernel Hilbert Space (RKHS) geometry. We first outline an exact computational routine, which is subsequently scaled for large datasets using randomized Nystrom approximations. Based on these foundations, we introduce the Kernel-SPV and Approximate Kernel-SPV algorithms for targeted subspace refinement via principal vectors. Simulation results validate our approach.

翻译：无限维库普曼算子的数据驱动近似依赖于有限维投影，所得模型的预测精度高度依赖于所选子空间的不变性。子空间剪枝通过系统性地剔除几何失准方向来增强这种不变性邻近性，这在形式上对应于子空间与其在算子作用下的像之间的最大主角。然而，现有技术主要局限于欧几里得设定。为弥补这一空白，本文提出了一种在再生核希尔伯特空间几何内计算主角与主向量的方法，以实现库普曼子空间剪枝。我们首先概述了精确计算流程，随后通过随机化奈斯特龙姆近似将其扩展至大规模数据集。基于这些基础，我们引入了Kernel-SPV与Approximate Kernel-SPV算法，通过主向量实现目标子空间精化。仿真结果验证了本方法的有效性。