Consistency of Interpolation with Laplace Kernels is a High-Dimensional Phenomenon
This addresses the theoretical understanding of interpolation phenomena in machine learning, particularly for researchers studying high-dimensional data and kernel methods, but is incremental as it builds on existing observations.
The paper demonstrates that minimum-norm interpolation with Laplace kernels fails to be consistent in constant input dimensions, regardless of kernel bandwidth selection, supporting empirical evidence that such interpolation generalizes well only in high-dimensional settings.
We show that minimum-norm 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 minimum-norm interpolation (that is, exact fit to training data) in RKHS generalizes well for some high-dimensional datasets, but not for low-dimensional ones.