We study the rank of sub-matrices arising out of kernel functions, $F(\pmb{x},\pmb{y}): \mathbb{R}^d \times \mathbb{R}^d \mapsto \mathbb{R}$, where $\pmb{x},\pmb{y} \in \mathbb{R}^d$, that have a singularity along $\pmb{x}=\pmb{y}$. Such kernel functions are frequently encountered in a wide range of applications such as $N$ body problems, Green's functions, integral equations, geostatistics, kriging, Gaussian processes, etc. One of the challenges in dealing with these kernel functions is that the corresponding matrix associated with these kernels is large and dense and thereby, the computational cost of matrix operations is high. In this article, we prove new theorems bounding the numerical rank of sub-matrices arising out of these kernel functions. Under reasonably mild assumptions, we prove that the rank of certain sub-matrices is rank-deficient in finite precision. This rank depends on the dimension of the ambient space and also on the type of interaction between the hyper-cubes containing the corresponding set of particles. This rank structure can be leveraged to reduce the computational cost of certain matrix operations such as matrix-vector products, solving linear systems, etc. We also present numerical results on the growth of rank of certain sub-matrices in $1$D, $2$D, $3$D and $4$D, which, not surprisingly, agrees with the theoretical results.
翻译:我们研究由内核函数产生的子矩阵的级别, $F (\ pmb{x},\ pmb{y} :\ mathbb{R ⁇ d\ times\ mathb{R ⁇ d\ mapsto\ mathb{R}R}$, 其中, $\ pmb{x}, pmb{y}\ in\ mathbb{R} 内, 与 $( pmb{x%b{y} ) 一起具有奇异性的 。 这种内核函数经常在一系列的应用中遇到, 例如 $N$( ) 体问题、 Green 函数、 整体方程式、 地理统计学、 krig、 高斯进程等。 处理这些内核函数的一个难题是, 与这些内核内核质相关的相应矩阵, 因此, 矩阵操作的计算成本很高 。 在此文章中, 我们证明新的内核内核数据组的内基体操作等级, 的内空数据级的数值等级 也必然会降低。