Common kernel ridge regression is expensive in memory allocation and computation time. This paper addresses low rank approximations and surrogates for kernel ridge regression, which bridge these difficulties. The fundamental contribution of the paper is a lower bound on the rank of the low dimensional approximation, which is required such that the prediction power remains reliable. The bound relates the effective dimension with the largest statistical leverage score. We characterize the effective dimension and its growth behavior with respect to the regularization parameter by involving the regularity of the kernel. This growth is demonstrated to be asymptotically logarithmic for suitably chosen kernels, justifying low-rank approximations as the Nystr\"om method.

翻译：普通核岭回归在内存分配和计算时间上代价高昂。本文针对核岭回归的低秩近似与替代方法展开研究，以克服上述困难。核心贡献在于给出了保证预测能力可靠所需的低维近似秩的下界。该界将有效维度与最大统计杠杆得分相关联。我们通过引入核的正则性，刻画了有效维度及其随正则化参数增长的行为。对于适当选取的核，该增长被证明具有渐进对数性质，从而验证了低秩近似（如Nyström方法）的合理性。