  Dimensionality reduction is in demand to reduce the complexity of solving
large-scale problems with data lying in latent low-dimensional structures in
machine learning and computer version. Motivated by such need, in this work we
study the Restricted Isometry Property (RIP) of Gaussian random projections for
low-dimensional subspaces in $\mathbb{R}^N$, and rigorously prove that the
projection Frobenius norm distance between any two subspaces spanned by the
projected data in $\mathbb{R}^n$ ($n<N$) remain almost the same as the distance
between the original subspaces with probability no less than $1 - {\rm
e}^{-\mathcal{O}(n)}$. Previously the well-known Johnson-Lindenstrauss (JL)
Lemma and RIP for sparse vectors have been the foundation of sparse signal
processing including Compressed Sensing. As an analogy to JL Lemma and RIP for
sparse vectors, this work allows the use of random projections to reduce the
ambient dimension with the theoretical guarantee that the distance between
subspaces after compression is well preserved.
