  We develop a general framework to significantly reduce the degree of
sum-of-squares proofs by introducing new variables. To illustrate the power of
this framework, we use it to speed up previous algorithms based on
sum-of-squares for two important estimation problems, clustering and robust
moment estimation. The resulting algorithms offer the same statistical
guarantees as the previous best algorithms but have significantly faster
running times. Roughly speaking, given a sample of $n$ points in dimension $d$,
our algorithms can exploit order-$\ell$ moments in time $d^{O(\ell)}\cdot
n^{O(1)}$, whereas a naive implementation requires time $(d\cdot n)^{O(\ell)}$.
Since for the aforementioned applications, the typical sample size is
$d^{\Theta(\ell)}$, our framework improves running times from $d^{O(\ell^2)}$
to $d^{O(\ell)}$.
