A well-recognized limitation of kernel learning is the requirement to handle
a kernel matrix, whose size is quadratic in the number of training examples.
Many methods have been proposed to reduce this computational cost, mostly by
using a subset of the kernel matrix entries, or some form of low-rank matrix
approximation, or a random projection method. In this paper, we study lower
bounds on the error attainable by such methods as a function of the number of
entries observed in the kernel matrix or the rank of an approximate kernel
matrix. We show that there are kernel learning problems where no such method
will lead to non-trivial computational savings. Our results also quantify how
the problem difficulty depends on parameters such as the nature of the loss
function, the regularization parameter, the norm of the desired predictor, and
the kernel matrix rank. Our results also suggest cases where more efficient
kernel learning might be possible.

本文主要研究降低核矩阵计算成本的方法，并针对观测的核矩阵条目数或近似核矩阵的秩，确定这些方法在错误达到下限的条件。研究结果表明损失函数，正则化参数，期望预测器的范数和核矩阵秩等参数与问题难度有关，同时提出了更高效的核学习可能的情况。