Rank of Matrices Arising out of Singular Kernel Functions

Kernel functions are frequently encountered in differential equations and machine learning applications. In this work, we study the rank of matrices arising out of the kernel function $K: X \times Y \mapsto \mathbb{R}$, where the sets $X, Y \in \mathbb{R}^d$ are hypercubes that share a boundary. The main contribution of this work is the analysis of the rank of such matrices where the particles (sources/targets) are arbitrarily distributed within these hypercubes. To our knowledge, this is the first work to formally investigate the rank of such matrices for an arbitrary distribution of particles. We model the arbitrary distribution of particles to arise from an underlying random distribution and obtain bounds on the expected rank and variance of the rank of the kernel matrix corresponding to various neighbor interactions. These bounds are useful for understanding the performance and complexity of hierarchical matrix algorithms (especially hierarchical matrices satisfying the weak-admissibility criterion) for an arbitrary distribution of particles. We also present numerical experiments in one-, two-, and three-dimensions, showing the expected rank growth and variance of the rank for different types of interactions. The numerical results, not surprisingly, align with our theoretical predictions.