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.

翻译：本文推导了通过截断特征分解（或奇异值分解）获得的核矩阵低秩逼近的逐项误差界。虽然该逼近在谱范数和Frobenius范数误差意义下的最优性已广为人知，但关于其各独立元素的统计行为却知之甚少。我们的误差界填补了这一空白。一个关键的技术创新是受随机矩阵理论启发，得到了核矩阵对应于小特征值的特征向量的去局域化结果。最后，我们通过对一系列合成及真实世界数据集的实证研究验证了理论。