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We show that minimum-norm interpolation in the Reproducing Kernel HilbertSpace corresponding to the Laplace kernel is not consistent if input dimensionis constant. The lower bound holds for any choice of kernel bandwidth, even ifselected based on data. The result supports the empirical observation thatminimum-norm interpolation (that is, exact fit to training data) in RKHSgeneralizes well for some high-dimensional datasets, but not forlow-dimensional ones.

Consistency of Interpolation with Laplace Kernels is a High-Dimensional Phenomenon