Properties of Bernoulli Random Projection for Embedding Subspaces

Abstract

In our early work [1], we proved that the affinity (distance) between two subspaces after Gaussian random projection will concentrate around an estimation with overwhelming probability. In this work, we extend the previous results to the case of Bernoulli random projection, which is essentially different from the Gaussian case because it lacks the property of rotation invariance. Specifically, we prove that the affinity between two subspaces after Bernoulli random projection will concentrate around an estimation with overwhelming probability. Our works (both this paper and [1]) are dramatically different from the previous works in RIP (Restricted Isometry Property), because we studied distance (affinity) between linear subspaces before and after random projection, while the previous works mainly studied the distance between points.