Using perturbed qr factorizations to solve linear least-squares problems

We propose and analyze a new tool to help solve sparse linear least-squares problems min_x \\Ax - b\\₂. Our method is based on a sparse QR factorization of a low-rank perturbation  of A. More precisely, we show that the R factor of  is an effective preconditioner for the least-squares problem min_x \\Ax - b\\₂, when solved using LSQR. We propose applications for the new technique. When A is rank deficient, we can add rows to ensure that the preconditioner is well conditioned without column pivoting. When A is sparse except for a few dense rows, we can drop these dense rows from A to obtain Â. Another application is solving an updated or downdated problem. If R is a good preconditioner for the original problem A, it is a good preconditioner for the updated/downdated problem Â. We can also solve what-if scenarios, where we want to find the solution if a column of the original matrix is changed/removed. We present a spectral theory that analyzes the generalized spectrum of the pencil (A≠ A, R≠ R) and analyze the applications.