It is crucial to learn the shared structures among functional predictors, as these structures characterize how predictor components exert common effects and, more generally, how predictors are homogeneously associated with the response. However, learning from multiple functional predictors is challenging because response-predictor dependencies may vary across representation dimensions and emerge at multiple resolutions, ranging from globally shared effects to predictor-specific effects. To address this issue, we propose a filtration-based shared structure learning framework for multiple functional predictors. The proposed framework organizes predictors through a hierarchical forest structure, in which shared and predictor-specific components are progressively identified from coarse to fine filtration layers. Building on this structure, we develop a filtration-based pursuit pipeline for shared structure discovery, together with a filtrated functional partial least squares method for shared component extraction and coefficient estimation under the learned shared structures. Simulation studies show that the proposed framework is able to recover the dominant coarse-to-fine organization of the underlying shared structures and yield improved prediction performance relative to competing methods. Applied to lower-limb angular kinematics, the proposed framework improves evaluation accuracy and reveals interpretable joint coordination patterns associated with aging. More broadly, it provides a new multiscale representation-learning perspective for complex data consisting of multiple multidimensional objects.

翻译：学习功能预测变量间的共享结构至关重要，因为这些结构表征了预测变量组分如何发挥共同效应，更一般地，预测变量如何与响应变量同质关联。然而，从多个功能预测变量中学习具有挑战性，因为响应-预测变量依赖关系可能随表征维度变化，并在从全局共享效应到预测变量特定效应的多个分辨率上显现。为解决该问题，我们提出一种基于过滤的共享结构学习框架用于多个功能预测变量。该框架通过层级森林结构组织预测变量，其中共享与预测变量特定组分从粗到细的过滤层中被逐步识别。基于此结构，我们开发了用于共享结构发现的过滤式追踪流程，并结合过滤式函数偏最小二乘法，在习得的共享结构下实现共享组件提取与系数估计。模拟研究表明，该框架能够恢复潜在共享结构的主导性由粗到细组织，并相较于竞争方法展现更优的预测性能。应用于下肢关节角度运动学数据时，该框架提升了评估准确性，并揭示了与衰老相关的可解释关节协调模式。更广泛而言，它为包含多个多维对象的复杂数据提供了新的多尺度表征学习视角。