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The leverage scores of a full-column rank matrix A are the squared row normsof any orthonormal basis for range(A). We show that corresponding leveragescores of two matrices A and A + \Delta A are close in the relative sense, ifthey have large magnitude and if all principal angles between the column spacesof A and A + \Delta A are small. We also show three classes of bounds that arebased on perturbation results of QR decompositions. They demonstrate thatrelative differences between individual leverage scores strongly depend on theparticular type of perturbation \Delta A. The bounds imply that the relativeaccuracy of an individual leverage score depends on: its magnitude and thetwo-norm condition of A, if \Delta A is a general perturbation; the two-normcondition number of A, if \Delta A is a perturbation with the same norm-wiserow-scaling as A; (to first order) neither condition number nor leverage scoremagnitude, if \Delta A is a component-wise row-scaled perturbation. Numericalexperiments confirm the qualitative and quantitative accuracy of our bounds.

Conditioning of Leverage Scores and Computation by QR Decomposition