2 research outputs found
Conditioning of Leverage Scores and Computation by QR Decomposition
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 + \Delta A are close in the relative sense, if
they have large magnitude and if all principal angles between the column spaces
of A and A + \Delta A 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 \Delta A. The bounds imply that the relative
accuracy of an individual leverage score depends on: its magnitude and the
two-norm condition of A, if \Delta A is a general perturbation; the two-norm
condition number of A, if \Delta A is a perturbation with the same norm-wise
row-scaling as A; (to first order) neither condition number nor leverage score
magnitude, if \Delta A is a component-wise row-scaled perturbation. Numerical
experiments confirm the qualitative and quantitative accuracy of our bounds.Comment: This version has been accepted to SIMAX but has not yet gone through
copy editin
Perturbation Analysis of the QR Factor R in the Context of LLL Lattice Basis Reduction
... \ud
computable notion of reduction of basis of a Euclidean lattice that is now commonly referred to as LLLreduction. The precise definition involves the R-factor of the QR factorisation of the basis matrix. A natural mean of speeding up the LLL reduction algorithm is to use a (floating-point) approximation to the R-factor. In the present article, we investigate the accuracy of the factor R of the QR factorisation of an LLL-reduced basis. The results we obtain should be very useful to devise LLL-type algorithms relying on floating-point approximations