AI 中文总结
该研究针对连续核函数产生的核矩阵，提出基于残差能量的框架构造低秩逼近，通过自适应选最优节点，有收敛保证，经实验验证其误差接近截断奇异值分解，鲁棒性强，确立了其作为经典交叉逼近技术连续对应方法的地位。
AI 中文摘要
我们提出了一个基于残差能量的框架，用于构造由连续核函数产生的核矩阵的低秩逼近。该方法在连续设置下运行，基于对枢轴节点（称为“最优节点”）的自适应选择，这些节点在每一步被选择以最小化残差能量。这导致残差核的一系列秩-1更新，并可自然地解释为自适应交叉逼近（ACA）的连续模拟。从理论角度看，我们表明残差核仍在紧算子类中，且逼近误差由残差能量精确表征。我们提供了收敛保证，表明该方法在对齐条件下产生单调误差减少，在实际动机假设下实现几何衰减。大量数值实验表明，该方法在一系列核函数上实现的逼近误差接近截断奇异值分解的误差。该方法在采样方面表现出强大的鲁棒性，在不同离散化下保持稳定性能。此外，连续残差能量与离散逼近误差之间的紧密一致性突出了公式的一致性。这些结果将所提出的方法确立为经典交叉逼近技术在理论上有依据、实践上有效的连续对应方法。
英文摘要
We propose a residual energy-based framework for constructing low-rank approximations of kernel matrices arising from continuous kernel functions. The method operates in a continuous setting and is based on the adaptive selection of pivot nodes, referred to as \emph{optimal nodes}, which are chosen to minimize the residual energy at each step. This leads to a sequence of rank-$1$ updates of the residual kernel and admits a natural interpretation as a continuous analog of Adaptive Cross Approximation (ACA). From a theoretical perspective, we show that the residual kernels remain in the class of compact operators and that the approximation error is exactly characterized by the residual energy. We provide convergence guarantees showing that the method yields monotonic error reduction under an alignment condition and achieves geometric decay under practically motivated assumptions. Extensive numerical experiments demonstrate that the proposed method achieves approximation errors close to those of the truncated singular value decomposition across a range of kernel functions. The method exhibits strong robustness with respect to sampling and maintains stable performance across different discretizations. Furthermore, the close agreement between the continuous residual energy and the discrete approximation error highlights the consistency of the formulation. These results establish the proposed approach as a theoretically grounded, practically effective continuous counterpart to classical cross-approximation techniques.
Comments16 pages, 6 figures, 1 table
