A Unified Theorem on SDP Rank Reduction

We consider the problem of finding a low–rank approximate solution to a system of linear equations in symmetric, positive semidefinite matrices, where the approximation quality of a solution is measured by its maximum relative deviation, both above and below, from the prescribed quantities. We show that a simple randomized polynomial–time procedure produces a low–rank solution that has provably good approximation qualities. Our result provides a unified treatment of and generalizes several well–known results in the literature. In particular, it contains as special cases the Johnson–Lindenstrauss lemma on dimensionality reduction, results on low–distortion embeddings into low–dimensional Euclidean space, and approximation results on certain quadratic optimization problems.