Conditioning of Leverage Scores and Computation by QR Decomposition

Abstract

The leverage scores of a full-column rank matrix A are the squared row norms of any orthonormal basis for range (A). We show that corresponding leverage scores of two matrices A and A + ΔA are close in the relative sense if they have large magnitude and if all principal angles between the column spaces of A and A + ΔA are small. We also show three classes of bounds that are based on perturbation results of QR decompositions. They demonstrate that relative differences between individual leverage scores strongly depend on the particular type of perturbation ΔA. The bounds imply that the relative accuracy of an individual leverage score depends on its magnitude and the two-norm condition of A if ΔA is a general perturbation; the two-norm condition number of A if ΔA is a perturbation with the same normwise row-scaling as A; (to first order) neither condition number nor leverage score magnitude if ΔA is a componentwise row-scaled perturbation. Numerical experiments confirm the qualitative and quantitative accuracy of our bounds.