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Low-rank approximations are popular methods to reduce the high computationalcost of algorithms involving large-scale kernel matrices. The success oflow-rank methods hinges on the matrix rank of the kernel matrix, and inpractice, these methods are effective even for high-dimensional datasets. Theirpractical success motivates our analysis of the function rank, an upper boundof 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 sumof separate products and provide explicit upper bounds on the function rank andthe $L_\infty$ error for such approximations. Our three main results are asfollows. First, for a fixed precision, the function rank of RBFs, in the worstcase, grows polynomially with the data dimension. Second, precise error boundsfor the low-rank approximations in the $L_\infty$ norm are derived in terms ofthe function smoothness and the domain diameters. Finally, a group pattern inthe magnitude of singular values for RBF kernel matrices is observed andanalyzed, and is explained by a grouping of the expansion terms in the kernel'slow-rank representation. Empirical results verify the theoretical results.

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