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 $d^{O(\ell)}$.

翻译：我们制定了一个总体框架,通过引入新的变量来大幅降低方块证据的总和程度。为了说明这个框架的力量,我们用它来加快基于两个重要估算问题、组合和稳健的瞬间估计的方块总和的先前算法。由此产生的算法提供了与以往最佳算法相同的统计保障,但运行速度要快得多。粗略地说,鉴于以美元为维度的零点样本,我们的算法可以利用美元时数的定值-美元,而天真的执行需要时间$(d\cdot n)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\