Restricted isometric compression of sparse datasets into low-dimensional varieties

This article extends the known restricted isometric projection of sparse datasets in Euclidean spaces $\mathbb{R}^N$ down into low-dimensional subspaces $\mathbb{R}^k, k \ll N,$ to the case of low-dimensional varieties $\mathcal{M} \subset \mathbb{R}^N,$ of codimension $N - k = \omega(N)$. Applications to structured/hierarchical datasets are considered.