论文标题
对具有歧管约束和复合二次惩罚的多任务功能线性回归模型的统一分析
A Unified Analysis of Multi-task Functional Linear Regression Models with Manifold Constraint and Composite Quadratic Penalty
论文作者
论文摘要
这项工作研究了多任务功能线性回归模型，其中协变量和未知回归系数（称为斜率函数）都是曲线。为了估算坡度函数，我们使用受惩罚的花样来平衡偏见，方差和计算复杂性。多任务学习的力量是通过在斜率函数上施加其他结构来实现的。我们提出了一个在样条系数矩阵上具有双重正则化的通用模型：i）矩阵歧管约束，ii）复合惩罚作为二次术语的总和。许多多任务学习方法可以视为该提出的模型的特殊情况，例如降低的模型和图形laplacian正规化模型。我们显示的综合惩罚会导致特定的规范，这有助于量化歧管曲率并确定歧管切线空间中相应的适当子集。然后将切线空间子集的复杂性桥接到通过通用链接的大地邻居的复杂性。获得统一收敛的上限，并专门应用于还原级别的模型和图形laplacian正则化模型。当我们改变模型参数的配置时，检查了估计器的相变行为。
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.
