Entrywise error bounds for low-rank approximations of kernel matrices
This work addresses a specific gap in kernel matrix approximation theory, providing incremental insights for researchers in machine learning and numerical analysis.
The paper tackles the problem of understanding entrywise error bounds for low-rank approximations of kernel matrices using truncated eigen-decomposition, deriving new bounds that address gaps in statistical behavior of individual entries, with validation on synthetic and real-world datasets.
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.