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.