Efficient error estimators for Generalized Nystr\"om

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 Nystr\"om decomposition, enabling accurate error estimation with no additional matrix accesses. In this work, we extend the leave-one-out framework to the generalized Nystr\"om 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.