A fast randomized algorithm for orthogonal projection

We describe an algorithm that, given any full-rank matrix A having fewer rows than columns, can rapidly compute the orthogonal projection of any vector onto the null space of A, as well as the orthogonal projection onto the row space of A, provided that both A and its adjoint A can be applied rapidly to arbitrary vectors. As an intermediate step, the algorithm solves the overdetermined linear least-squares regression involving A (and so can be used for this, too). The basis of the algorithm is an obvious but numerically unstable scheme; suitable use of a preconditioner yields numerical stability. We generate the preconditioner rapidly via a randomized procedure that succeeds with extremely high probability. In many circumstances, the method can accelerate interior-point methods for convex optimization, such as linear programming (Ming Gu, personal communication).