  We present a relation between sparsity and non-Euclidean isomorphic
embeddings. We introduce a general restricted isomorphism property and show how
it enables to construct embeddings of $\ell_p^n$, $p > 0$, into various type of
Banach or quasi-Banach spaces. In particular, for $0 <r < p<2$ with $r \le 1$,
we construct a family of operators that embed $\ell_p^n$ into
$\ell_r^{(1+\eta)n}$, with optimal polynomial bounds in $\eta >0$.
