Predictive Nyström method for kernel methods

Nyström method is a widely used matrix approximation method for scaling up kernel methods, and existing sampling strategies for Nyström method are proposed to improve the matrix approximation accuracy, but leaving approximation independent of learning, which can result in poor predictive performance of kernel methods. In this paper, we propose a novel predictive sampling strategy (PRESS) for Nyström method that guarantees the predictive performance of kernel methods. PRESS adaptively updates the sampling distribution via the discrepancy between approximate and accurate solutions of kernel methods caused by kernel matrix approximation, and samples informative columns from the kernel matrix according to the sampling distribution to reduce the predictive performance loss of kernel methods. We prove upper error bounds on the approximate solutions of kernel methods produced by Nyström method with PRESS, whose convergence shows that approximate solutions of kernel methods are identical to accurate ones for large enough samples. Experimental results indicate that integrating learning into approximation is necessary for delivering better predictive performance, and PRESS significantly outperforms existing sampling strategies while preserving low computational cost.