# On the numerical rank of radial basis function kernels in high dimension

## TL;DR

This paper analyzes the function rank of radial basis function kernels in high dimensions, providing explicit bounds and insights into their low-rank approximations, which are crucial for efficient large-scale kernel methods.

## Contribution

It offers explicit upper bounds on the function rank of RBF kernels, analyzes the growth with data dimension, and explains the singular value grouping observed in kernel matrices.

## Key findings

- Function rank grows polynomially with data dimension in worst case.
- Derived error bounds depend on smoothness and domain size.
- Observed singular value grouping explained by low-rank expansion terms.

## Abstract

Low-rank approximations are popular methods to reduce the high computational cost of algorithms involving large-scale kernel matrices. The success of low-rank methods hinges on the matrix rank of the kernel matrix, and in practice, these methods are effective even for high-dimensional datasets. Their practical success motivates our analysis of the function rank, an upper bound of the matrix rank. In this paper, we consider radial basis functions (RBF), approximate the RBF kernel with a low-rank representation that is a finite sum of separate products and provide explicit upper bounds on the function rank and the $L_\infty$ error for such approximations. Our three main results are as follows. First, for a fixed precision, the function rank of RBFs, in the worst case, grows polynomially with the data dimension. Second, precise error bounds for the low-rank approximations in the $L_\infty$ norm are derived in terms of the function smoothness and the domain diameters. Finally, a group pattern in the magnitude of singular values for RBF kernel matrices is observed and analyzed, and is explained by a grouping of the expansion terms in the kernel's low-rank representation. Empirical results verify the theoretical results.