发表机构
University of Calgary; York University; Simon Fraser University（卡尔加里大学; 约克大学; 西蒙弗雷泽大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对函数型回归中系数曲面支撑结构复杂的问题，提出FAIR方法，利用自注意力学习效应自适应邻域，结合稀疏和平滑惩罚，实现更准确的支撑恢复与预测。
AI 中文摘要
在函数对函数回归中，系数曲面 $\eta(s,t)$ 可能表现出复杂的支撑结构——从局部斑块到全局模式，如不连通区域、带状或环形结构——其中效应相似性与欧几里得邻近性并不一致。依赖固定基展开的投影方法可能掩盖此类结构，而直接平滑方法则面临对曲面及其边界过度平滑的风险。我们提出函数型注意力可解释回归（FAIR），该方法通过坐标特征直接表示 $\eta(s,t)$，并利用自注意力学习效应自适应邻域，从而在局部和全局尺度上实现信息共享。一个标量压缩网络将这些学习到的表示映射到系数曲面。在这些邻域上施加的稀疏性和平滑性惩罚促进了具有连贯边界的局部支撑。我们建立了与张量积样条空间的筛等价性，并推导了收敛速率。模拟实验以及对海洋学和水文学数据的应用表明，FAIR 在恢复支撑几何结构方面比现有方法更准确，同时在预测性能上更优，尤其是在稀疏采样条件下。
英文摘要
In function-on-function regression, the coefficient surface $β(s,t)$ may exhibit complex support structure---from localized patches to global patterns such as disconnected regions, bands, or rings---where effect similarity does not align with Euclidean proximity. Projection-based methods that rely on fixed basis expansions can obscure such structure, while direct smoothing approaches risk oversmoothing the surface and its boundaries. We propose Functional Attentive Interpretable Regression (FAIR), which represents $β(s,t)$ directly through coordinate features and uses self-attention to learn effect-adaptive neighborhoods, enabling information sharing at both local and global scales. A scalar compression network maps these learned representations to the coefficient surface. Sparsity and smoothness penalties applied over these neighborhoods promote localized support with coherent boundaries. We establish a sieve equivalence to tensor-product spline spaces and derive convergence rates. Simulations and applications to oceanographic and hydrological data demonstrate that FAIR recovers support geometry more accurately than existing methods while achieving superior prediction, particularly under sparse sampling.
