  The $l_2$ flattening lemma of Johnson and Lindenstrauss [JL84] is a powerful
tool for dimension reduction. It has been conjectured that the target dimension
bounds can be refined and bounded in terms of the intrinsic dimensionality of
the data set (for example, the doubling dimension). One such problem was
proposed by Lang and Plaut [LP01] (see also
[GKL03,MatousekProblems07,ABN08,CGT10]), and is still open. We prove another
result in this line of work:
  The snowflake metric $d^{1/2}$ of a doubling set $S \subset l_2$ embeds with
constant distortion into $l_2^D$, for dimension $D$ that depends solely on the
doubling constant of the metric. In fact, the distortion can be made
arbitrarily close to 1, and the target dimension is polylogarithmic in the
doubling constant. Our techniques are robust and extend to the more difficult
spaces $l_1$ and $l_\infty$, although the dimension bounds here are
quantitatively inferior than those for $l_2$.
