On the numerical rank of radial basis function kernels in high dimension
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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, and in practice, these methods are effective even for high-dimensional datasets. The practical success has elicited the theoretical analysis of the function rank in this paper, which is an upper bound of the matrix rank. The concept of function rank will be introduced to define the number of terms in the minimal separate form of a kernel function. We consider radial basis functions (RBF) in particular, and 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_∞ error for such approximation. 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_∞ norm are derived in terms of the function smoothness and the domain diameters. And last, 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.
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