Subspace clustering with random transformation
It is possible that choosing $R$ according to a uniform distribution with zero mean constitutes a Johnson–Lindenstrauss random projection (i.e., it approximately preserves distances). See for example a paper by Achlioptas which suggests three distributions that definitely work: a zero-mean Gaussian, a random $\pm 1$, and a specific distribution on $\{0,\pm 1\}$. It is probable that using the methods in the paper, the case of the uniform distribution can also be analyzed.