Consistency of Interpolation with Laplace Kernels is a HighDimensional Phenomenon
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Proceedings of the ThirtySecond Conference on Learning Theory, PMLR 99:25952623, 2019.
Abstract
We show that minimumnorm interpolation in the Reproducing Kernel Hilbert Space corresponding to the Laplace kernel is not consistent if input dimension is constant. The lower bound holds for any choice of kernel bandwidth, even if selected based on data. The result supports the empirical observation that minimumnorm interpolation (that is, exact fit to training data) in RKHS generalizes well for some highdimensional datasets, but not for lowdimensional ones.
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