The $2 \rightarrow q$ norm of a matrix $X \in \mathbb{R}^{n \times d}$ is defined as $\lVert X \rVert_{2 \rightarrow q} = \sup_{\lVert v \rVert_2 = 1} \lVert Xv \rVert_q$. We give polynomial-time multiplicative approximation algorithms for this norm when $q > 2$ (i.e. in the hypercontractive setting). This problem either directly captures or is closely related to long-standing open problems in combinatorial optimization and hardness of approximation (e.g. Small Set Expansion), quantum information (e.g. Best Separable State), and algorithmic statistics. Very little is known about what approximation factors we can achieve for this problem in polynomial time, even though such approximations have significant downstream consequences. Barak, Brandão, Harrow, Kelner, Steurer, and Zhou showed that no polynomial-time algorithm can achieve an approximation factor better than $2^{\sqrt{\log n}}$, assuming the Exponential Time Hypothesis (FOCS'12). On the other hand, a simple spectral algorithm gives a $d^{1/4}$-approximation as a baseline. We give, to the best of our knowledge, the first polynomial-time approximation algorithm beating this baseline by polynomial factors. For the important special case of $q = 4$ it achieves a $d^{1/8}$-approximation. All previous algorithms required additional assumptions on $X$, or only surpassed the baseline for small values of $n$. Moreover, we construct sum-of-squares certificates for the $2 \rightarrow q$ norm. This directly implies improved algorithms for robust mean and covariance estimation, robust regression, and clustering, when the data only satisfies a bound on its $q$-th moment.

翻译：矩阵$X \in \mathbb{R}^{n \times d}$的$2 \rightarrow q$范数定义为$\lVert X \rVert_{2 \rightarrow q} = \sup_{\lVert v \rVert_2 = 1} \lVert Xv \rVert_q$。我们针对$q > 2$的情形（即超压缩设定）给出了该范数的多项式时间乘性逼近算法。该问题要么直接对应、要么与组合优化和近似难度（例如小集扩张）、量子信息（例如最佳可分离态）以及算法统计学中长期存在的开放问题密切相关。关于在多项式时间内对该问题能达到何种逼近因子，目前所知甚少，尽管此类逼近具有重要的下游应用。Barak、Brandão、Harrow、Kelner、Steurer和Zhou证明，假设指数时间假说成立（FOCS'12），则不存在多项式时间算法能实现优于$2^{\sqrt{\log n}}$的逼近因子。另一方面，作为基线，一个简单的谱算法可达到$d^{1/4}$逼近。据我们所知，我们首次给出了多项式时间逼近算法，能够以多项式因子超越该基线。对于$q = 4$这一重要特例，该算法实现了$d^{1/8}$逼近。此前所有算法要么需要对$X$施加额外假设，要么仅能在$n$较小时超越基线。此外，我们为$2 \rightarrow q$范数构造了平方和证书。当数据仅满足$q$阶矩有界条件时，这直接改进了鲁棒均值和协方差估计、鲁棒回归以及聚类问题的算法。