Efficient error estimators for Generalized Nyström

Abstract: Randomized algorithms in numerical linear algebra have proven to be effective in ameliorating issues of scalability when working with large matrices, efficiently producing accurate low-rank approximations. A key remaining challenge, however, is to efficiently assess the approximation accuracy of randomized methods without additional expensive matrix accesses. Recent work has addressed this issue by deriving fast leave-one-out error estimators for the randomized SVD and Nyström decomposition, enabling accurate error estimation with no additional matrix accesses. In this work, we extend the leave-one-out framework to the generalized Nyström decomposition, an approach that can be applied to general rectangular matrices. We do this by deriving three new leave-one-out error estimators and validating their effectiveness through numerical experiments.