AI 中文总结
该研究提出基于样本的框架求解条件正定核与通用克里金法的鞍点系统，通过约简实现O(N log N)复杂度，经三类应用验证有效性。
AI 中文摘要
我们提出一种基于样本（samplet）的框架，用于高效求解由条件正定（CPD）核近似（尤其是通用克里金法）产生的鞍点系统。样本的消失矩性质以及对应离散正交多项式的相关尺度分布的特殊结构，为这些鞍点系统提供了数值上有利的表示。具体而言，它们可将不定鞍点系统自然地约简为用于细节系数的（大型）线性系统和用于多项式系数的小型三角系统。我们推导了贝波-列维（Beppo-Levi）空间中多调和样条近似的误差界，并证明细节系数恰好张成CPD核为正定的子空间，使得约简后的块对称矩阵为正定。鉴于渐近光滑核的样本变换核矩阵的拟稀疏性，所提方法在鞍点系统的组装和存储上达到O(N log N)的复杂度。约简后的系统可通过稀疏乔列斯基分解高效求解。我们用三个应用示例说明该框架：广义协方差下的高斯过程回归、基于样本压缩薄板样条的地标图像配准，以及三维网格变形。
英文摘要
We present a samplet-based framework for the efficient numerical solution of saddle-point systems arising from conditionally positive definite (CPD) kernel approximation in general and universal Kriging in particular. The vanishing moment property of samplets as well as the particular structure of the associated scaling distributions, which correspond to discrete orthogonal polynomials, allow for a numerically favorable representation of these saddle-point systems. Concretely, they enable a natural null-space reduction of the indefinite saddle-point system to a (large) linear system for the detail coefficients and a small triangular system for the polynomial coefficients. We derive error bounds for the approximation by polyharmonic splines in Beppo-Levi spaces and show that the detail coefficients span precisely the subspace on which the CPD kernel is positive definite, rendering the reduced block symmetric positive definite. In view of the quasi-sparsity of the samplet-transformed kernel matrix for asymptotically smooth kernels, the resulting method achieves O(N log N) cost for the assembly and the storage of the saddle-point system. The reduced system can efficiently be solved by a sparse Cholesky factorization. We illustrate the framework with three applications, namely Gaussian process regression with generalized covariances, landmark-based image registration via samplet-compressed thin plate splines, and three-dimensional mesh deformation.
