利用覆盖网估计算子范数
Estimating operator norms using covering nets
摘要：
我们提出了几个多项式和拟多项式时间的近似计划的一大类广义算子规范。特殊情况包括矩阵的$2\rightarrow q$范数为$q>2$，支持功能的一组可分离的量子态，找到最少的噪音输出的纠缠打破量子信道，并近似内射张量范数为两个Banach空间之间的映射，其因子分解范数通过$\ell_1^n$是有界的。这些复制，并在某些情况下，提高了以前的算法的性能，由Brandeono-Christandl-Yard和后续工作，这是基于平方和层次结构，其分析使用的技术从量子信息，如一夫一妻制原则的纠缠。相比之下，我们的算法是基于蛮力枚举仔细选择的覆盖网。这些方法的优点是使用更少的内存，具有更简单的证明，并为问题提供新的几何见解。Shi-Wu和Barak-Kelner-Steurer也提出了类似问题的基于网络的算法，但在每种情况下，运行时间都是某个矩阵秩的指数。我们通过使用$\ell_1$空间中存在的小得多的网来实现多项式或准多项式运行时。这一原则已被用于学习理论，在那里它被称为莫雷的经验方法。
英文摘要：
We present several polynomial- and quasipolynomial-time approximation schemes for a large class of generalized operator norms. Special cases include the $2\rightarrow q$ norm of matrices for $q>2$, the support function of the set of separable quantum states, finding the least noisy output of entanglement-breaking quantum channels, and approximating the injective tensor norm for a map between two Banach spaces whose factorization norm through $\ell_1^n$ is bounded. These reproduce and in some cases improve upon the performance of previous algorithms by Brandão-Christandl-Yard and followup work, which were based on the Sum-of-Squares hierarchy and whose analysis used techniques from quantum information such as the monogamy principle of entanglement. Our algorithms, by contrast, are based on brute force enumeration over carefully chosen covering nets. These have the advantage of using less memory, having much simpler proofs and giving new geometric insights into the problem. Net-based algorithms for similar problems were also presented by Shi-Wu and Barak-Kelner-Steurer, but in each case with a run-time that is exponential in the rank of some matrix. We achieve polynomial or quasipolynomial runtimes by using the much smaller nets that exist in $\ell_1$ spaces. This principle has been used in learning theory, where it is known as Maurey's empirical method.
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