使用截断Demmler-Reinsch基的真正最优低秩薄板样条平滑
Truly optimal low rank thin plate spline smoothing using a truncated Demmler-Reinsch basis
AI总结：
针对薄板样条计算成本高及TPRS近似最优性不足的问题，提出基于截断Demmler-Reinsch基的低秩近似，证明其最优收敛速率并给出高效算法，性能与TPRS相当且更具优势。
AI中文摘要：
薄板样条是极具吸引力的平滑器。然而，其计算成本为三次方，严重限制了其在实际中的应用。作为补救措施，Wood (2003) 提出了薄板回归样条（TPRS），它提供了低秩近似。TPRS近似的关键步骤是对径向基函数（RBF）设计矩阵进行截断特征分解。然而，正如Wood (2003) 所述，TPRS近似的最优性是稍弱的。这是因为RBF系数受到正交性约束，而TPRS近似仅在忽略这些约束时才是最优的。为解决这一缺陷，我们提出了一种略有不同的低秩近似。所提出的近似基于截断Demmler-Reinsch基（TDRB），该基在Frobenius范数和谱范数意义上提供了平滑矩阵的最佳低秩近似。我们证明了TDRB平滑器达到了最优收敛速率，并提出了一种高效的构造算法。该算法基于等效贝叶斯平滑先验的截断Karhunen-Loève（KL）展开，其计算成本与TPRS相同。我们通过模拟和真实数据示例展示了我们方法的适用性。我们发现其性能与TPRS非常相似，但所提出的方法具有一些优势。
英文摘要：
Thin plate splines are highly attractive smoothers. However, they have cubic computational cost, which severely limits their use in practice. As a remedy, Wood (2003) suggested thin plate regression splines (TPRS), which provide a low rank approximation. The key step of the TPRS approximation is a truncated eigendecomposition of the radial basis function (RBF) design matrix. However, as Wood (2003) writes, the optimality of the TPRS approximation is a slightly weak one. This is because the RBF coefficients are subject to orthogonality constraints and the TPRS approximation is only optimal if these constraints are ignored. To address this shortcoming, we suggest a slightly different low rank approximation. The suggested approximation is based on a truncated Demmler-Reinsch basis (TDRB), which provides a best low rank approximation of the smoother matrix in terms of Frobenius and spectral norm. We prove that the TDRB smoother achieves the optimal rate of convergence and suggest an efficient algorithm for its construction. This algorithm is based on a truncated Karhunen-Loève (KL) expansion of the equivalent Bayesian smoothness prior and it has the same computational cost as required for TPRS. We demonstrate the applicabilty of our approach through simulations and a real data example. We find that the performance is very similar to that of TPRS but the suggested approach has some advantages.
