On the numerical rank of radial basis function kernel matrices in high dimension
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Low-rank approximations are popular techniques to reduce the high computational cost of large-scale kernel matrices, which are of significant interest in many applications. The success of low-rank methods hinges on the matrix rank, and in practice, these methods are effective even for high-dimensional datasets. The practical success has elicited the theoretical analysis of the rank in this paper. We will consider radial basis functions (RBF) and present theorems on the rank and error bounds. Our three main results are as follows. First, the rank of RBFs grows polynomially with the data dimension, in the worst case; second, precise approximation error bounds in terms of function properties and data structures are derived; and last, a group pattern in the decay of singular values for RBF kernel matrices is analyzed, and is explained by a grouping of the expansion terms. Empirical results verify and illustrate the theoretical results.
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