Principal angles preserving property of Gaussian random projection for subspaces

With the complexity of solving large-scale problems increasing, dimension reduction appears more and more important. In the early work, we find that with high probability the affinity and distance between two subspaces will be concentrated around their estimates after Gaussian random projection. When the ambient dimension after projection is sufficiently large, the affinity and distance between two subspaces almost remain unchanged after projection. In this paper, we extend the above results to principal angles (or canonical angles) to reveal the relationship between two subspaces from a more fundamental viewpoint. We theoretically prove that each principal angle will also be concentrated around its estimate with high probability. When the ambient dimension after projection is sufficiently large, it also remains almost unchanged. Numerical simulation verifies the theoretical work.