This work studies the multi-task functional linear regression models where both the covariates and the unknown regression coefficients (called slope functions) are curves. For slope function estimation, we employ penalized splines to balance bias, variance, and computational complexity. The power of multi-task learning is brought in by imposing additional structures over the slope functions. We propose a general model with double regularization over the spline coefficient matrix: i) a matrix manifold constraint, and ii) a composite penalty as a summation of quadratic terms. Many multi-task learning approaches can be treated as special cases of this proposed model, such as a reduced-rank model and a graph Laplacian regularized model. We show the composite penalty induces a specific norm, which helps to quantify the manifold curvature and determine the corresponding proper subset in the manifold tangent space. The complexity of tangent space subset is then bridged to the complexity of geodesic neighbor via generic chaining. A unified convergence upper bound is obtained and specifically applied to the reduced-rank model and the graph Laplacian regularized model. The phase transition behaviors for the estimators are examined as we vary the configurations of model parameters.

翻译：本文研究多任务泛函线性回归模型，其中协变量和未知回归系数（称为斜率函数）均为曲线。在斜率函数估计中，我们采用惩罚样条来平衡偏差、方差和计算复杂度。通过为斜率函数施加额外结构，引入多任务学习的优势。我们提出一种对样条系数矩阵进行双重正则化的通用模型：i) 矩阵流形约束，ii) 由二次项求和构成的复合惩罚项。诸多多任务学习方法（如降秩模型和图拉普拉斯正则化模型）均可视为该模型的特例。研究表明，复合惩罚项可诱导出特定范数，该范数有助于量化流形曲率并确定流形切空间中对应的适当子集。切空间子集的复杂度通过广义链式法则与测地邻域的复杂度建立关联。我们获得了统一的收敛上界，并将其具体应用于降秩模型和图拉普拉斯正则化模型。通过改变模型参数配置，进一步研究了估计量的相变行为。