Rank of Matrices Arising out of Singular Kernel Functions

TL;DR

This work studies the rank properties of kernel matrices $K$ formed from interactions between source and target domains when the underlying particle locations are randomly distributed. By modeling the distributions as uniform i.i.d. samples and using a hierarchical subdivision with a random-rank framework, the authors derive explicit bounds on the expected rank $\mathbb{E}[\mathcal{R}]$ and its variance $\mathrm{Var}(\mathcal{R})$ for all neighbor interaction types in $d$ dimensions, showing $\mathbb{E}[\mathcal{R}] = \mathcal{O}(p\log_{2^d}(n))$ for vertex-sharing ($d'=0$) and $\mathbb{E}[\mathcal{R}] = \mathcal{O}(p n^{d'/d})$ for $d'$-dimensional surface-sharing, with corresponding variance bounds. The theory is supported by numerical experiments in 1D–3D across multiple kernels, confirming the predicted rank growth and variance behavior and implying robustness of hierarchical, weakly admissible matrix methods under realistic, random particle distributions. The results provide practical guarantees for the efficiency of low-rank approximations in large kernels arising in PDEs, integral equations, and related areas. A notable open question is whether the vertex-sharing rank $\mathcal{R}$ exhibits Gaussian behavior in the asymptotic limit.

Abstract

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.

Figures (11)

• Figure 13: Sub-division of the source $Y=[a,b]$ up to the level $\kappa$.
• Figure 14: Vertex-sharing interaction.
• Figure 15: Edge-sharing interaction.
• Figure 17: Vertex-sharing interaction.
• Figure 18: Edge-sharing interaction.

Theorems & Definitions (26)

• Definition 2.1
• Definition 2.2
• Remark 2.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 3.1
• Theorem 3.2
• Remark 4.1
• Lemma 4.1