Estimating singular functions of kernel cross-covariance operators: An investigation of the Nyström method

We investigate the Nyström method as an efficient means of overcoming the computational bottleneck inherent in estimating the singular functions of kernel cross-covariance operators, which play a central role in tasks such as covariate shift correction and multi-view learning. We present a Nyström-type approximation of the kernel cross-covariance operator, and establish its convergence rate. Furthermore, we derive a novel bound on the weighted sum of squared estimation errors of all associated singular functions, providing tighter control than traditional bounds that treat each error individually. Our theoretical analysis reveals that the Nyström-based singular function estimators attain the same statistical accuracy as their full empirical counterparts, while offering significant computational savings. Numerical experiments further confirm the practical effectiveness of the proposed approach.