  The Johnson-Lindenstrauss (JL) Lemma introduced the concept of dimension
reduction via a random linear map, which has become a fundamental technique in
many computational settings. For a set of $n$ points in $\mathbb{R}^d$ and any
fixed $\epsilon>0$, it reduces the dimension $d$ to $O(\log n)$ while
preserving, with high probability, all the pairwise Euclidean distances within
factor $1+\epsilon$. Perhaps surprisingly, the target dimension can be lower if
one only wishes to preserve the optimal value of a certain problem on the
pointset, e.g., Euclidean max-cut or $k$-means. However, for some notorious
problems, like diameter (aka furthest pair), dimension reduction via the JL map
to below $O(\log n)$ does not preserve the optimal value within factor
$1+\epsilon$.
  We propose to focus on another regime, of \emph{moderate dimension
reduction}, where a problem's value is preserved within factor $\alpha>1$ using
target dimension $\tfrac{\log n}{poly(\alpha)}$. We establish the viability of
this approach and show that the famous $k$-center problem is
$\alpha$-approximated when reducing to dimension $O(\tfrac{\log
n}{\alpha^2}+\log k)$. Along the way, we address the diameter problem via the
special case $k=1$. Our result extends to several important variants of
$k$-center (with outliers, capacities, or fairness constraints), and the bound
improves further with the input's doubling dimension.
  While our $poly(\alpha)$-factor improvement in the dimension may seem small,
it actually has significant implications for streaming algorithms, and easily
yields an algorithm for $k$-center in dynamic geometric streams, that achieves
$O(\alpha)$-approximation using space $poly(kdn^{1/\alpha^2})$. This is the
first algorithm to beat $O(n)$ space in high dimension $d$, as all previous
algorithms require space at least $\exp(d)$. Furthermore, it extends to the
$k$-center variants mentioned above.
