Numerical stability of the Rokhlin–Tygert sketch-based preconditioning for least-squares

Determine the numerical stability of the Rokhlin–Tygert randomized sketch-based preconditioning method for overdetermined least-squares problems, specifically whether solving the preconditioned problem with LSQR using the factor R from a subspace-embedded QR factorization of ΦA is backward stable in floating-point arithmetic, and derive rigorous error bounds or identify conditions under which stability is guaranteed.

Background

The notes present a randomized preconditioning strategy for overdetermined least-squares problems in which a fast subspace embedding Φ is applied to the tall matrix A, followed by a QR factorization ΦA = QR. Using the triangular factor R as a right preconditioner transforms the original least-squares problem into one with a very well-conditioned matrix AR^{-1}, enabling efficient solution via LSQR.

While the approach offers strong theoretical conditioning guarantees for the preconditioned matrix and reduced computational cost, the authors explicitly point out that the numerical stability of the full algorithm in floating-point arithmetic is not fully resolved and references recent work addressing this question.