Low-rank approximations of large kernel matrices are ubiquitous in machine learning, particularly for scaling Gaussian Processes to massive datasets. The Pivoted Cholesky decomposition is a standard tool for this task, offering a computationally efficient, greedy low-rank approximation. While its algebraic properties are well-documented in numerical linear algebra, its geometric intuition within the context of kernel methods often remains obscure. In this note, we elucidate the geometric interpretation of the algorithm within the Reproducing Kernel Hilbert Space (RKHS). We demonstrate that the pivotal selection step is mathematically equivalent to Farthest Point Sampling (FPS) using the kernel metric, and that the Cholesky factor construction is an implicit Gram-Schmidt orthogonalization. We provide a concise derivation and a minimalist Python implementation to bridge the gap between theory and practice.

翻译：大规模核矩阵的低秩近似在机器学习中无处不在，特别是在将高斯过程扩展至海量数据集时。枢轴Cholesky分解是完成此任务的标准工具，它提供了一种计算高效、贪婪的低秩近似方法。虽然其代数性质在数值线性代数中已有详尽记载，但在核方法背景下的几何直观往往仍不清晰。本文旨在阐明该算法在再生核希尔伯特空间（RKHS）中的几何解释。我们证明：枢轴选择步骤在数学上等价于使用核度量的最远点采样（FPS），而Cholesky因子构建则是一种隐式的Gram-Schmidt正交化过程。我们提供了简洁的推导和极简的Python实现，以弥合理论与实践之间的鸿沟。