  In this paper, we derive entrywise error bounds for low-rank approximations
of kernel matrices obtained using the truncated eigen-decomposition (or
singular value decomposition). While this approximation is well-known to be
optimal with respect to the spectral and Frobenius norm error, little is known
about the statistical behaviour of individual entries. Our error bounds fill
this gap. A key technical innovation is a delocalisation result for the
eigenvectors of the kernel matrix corresponding to small eigenvalues, which
takes inspiration from the field of Random Matrix Theory. Finally, we validate
our theory with an empirical study of a collection of synthetic and real-world
datasets.
