We illustrate relationships between classical kernel-based dimensionality reduction techniques and eigendecompositions of empirical estimates of reproducing kernel Hilbert space (RKHS) operators associated with dynamical systems. In particular, we show that kernel canonical correlation analysis (CCA) can be interpreted in terms of kernel transfer operators and that coherent sets of particle trajectories can be computed by applying kernel CCA to Lagrangian data. We demonstrate the efficiency of this approach with several examples, namely the well-known Bickley jet, ocean drifter data, and a molecular dynamics problem with a time-dependent potential. Furthermore, we propose a straightforward generalization of dynamic mode decomposition (DMD) called coherent mode decomposition (CMD).
翻译:我们举例说明了传统内核基维度减少技术与复制与动态系统相关的内核Hilbert空间(RKHS)操作员的经验估计的微数分解法之间的关系,特别是,我们表明内核相关分析可被解释为内核转移操作员,通过对Lagrangian数据应用内核分解法,可以计算出一致的粒子轨迹。我们通过几个例子,即众所周知的Bickley喷气机、海洋漂流数据以及具有时间性潜力的分子动态问题,证明了这种方法的效率。此外,我们建议直接概括被称为一致模式分解法的动态模式(DMD)的典型化。