矩阵$p\rightarrow q$范数的不可逼近性
Inapproximability of Matrix $p\rightarrow q$ Norms
摘要：
本文研究了矩阵$A \in R^{m \times n}$的$p\rightarrow q$范数的计算问题，定义为\[ \|A\|_{p\rightarrow q} ~:=~ \max_{x \,\in\, R^n \setminus \{0\}} \frac{\|Ax\|_q}{\|x\|_p} \]。这个问题推广了矩阵的谱范数（$p=q=2$）和Grothendieck问题（$p=\infty$，$q=1$），并在各种领域得到了广泛的研究。当$p \geq q$，问题表现出二分法：常数因子近似算法是已知的，如果$2 \in [q,p]$，和问题是很难近似在几乎多项式的因素时，$2 \notin [q,p]$。当$p < q$（被称为\emph{hypercontractive norms}）时的状态对于各种应用特别重要，但很少被很好地理解。[Barak et al，STOC'12]研究了$p = 2$和$q > 2$的情况，他给出了问题的承诺版本的次指数算法（捕获小集合扩展），并证明了基于指数时间假设的近似结果的硬度。然而，对于任何$p < q$，这些问题都没有NP-困难的近似。我们研究了在上述两种情况下，近似矩阵范数的硬度，并证明了以下结果：-我们表明，对于任何$1< p < q < \infty$与$2 \notin [p,q]$，$\|A\|_{p\rightarrow q}$是很难近似在$2^{O(\log^{1-ε}\!n)}$假设$NP \not\subseteq BPTIME(2^{\log^{O(1)}\!n})$。这表明，与$p \geq q$的情况类似，当$2$不在$p$和$q$之间时，高收缩设置可能是定性不同的。- 对于所有的$p \geq q$与$2 \in [q,p]$，我们表明$\|A\|_{p\rightarrow q}$是很难近似的任何因素比$1/(γ_{p^*} \cdot γ_q)$，其中对于任何$r$，$γ_r$表示的$r^{th}$规范的高斯，和$p^*$是$p$的对偶规范。
英文摘要：
We study the problem of computing the $p\rightarrow q$ norm of a matrix $A \in R^{m \times n}$, defined as \[ \|A\|_{p\rightarrow q} ~:=~ \max_{x \,\in\, R^n \setminus \{0\}} \frac{\|Ax\|_q}{\|x\|_p} \] This problem generalizes the spectral norm of a matrix ($p=q=2$) and the Grothendieck problem ($p=\infty$, $q=1$), and has been widely studied in various regimes. When $p \geq q$, the problem exhibits a dichotomy: constant factor approximation algorithms are known if $2 \in [q,p]$, and the problem is hard to approximate within almost polynomial factors when $2 \notin [q,p]$. The regime when $p < q$, known as \emph{hypercontractive norms}, is particularly significant for various applications but much less well understood. The case with $p = 2$ and $q > 2$ was studied by [Barak et al, STOC'12] who gave sub-exponential algorithms for a promise version of the problem (which captures small-set expansion) and also proved hardness of approximation results based on the Exponential Time Hypothesis. However, no NP-hardness of approximation is known for these problems for any $p < q$. We study the hardness of approximating matrix norms in both the above cases and prove the following results: - We show that for any $1< p < q < \infty$ with $2 \notin [p,q]$, $\|A\|_{p\rightarrow q}$ is hard to approximate within $2^{O(\log^{1-ε}\!n)}$ assuming $NP \not\subseteq BPTIME(2^{\log^{O(1)}\!n})$. This suggests that, similar to the case of $p \geq q$, the hypercontractive setting may be qualitatively different when $2$ does not lie between $p$ and $q$. - For all $p \geq q$ with $2 \in [q,p]$, we show $\|A\|_{p\rightarrow q}$ is hard to approximate within any factor than $1/(γ_{p^*} \cdot γ_q)$, where for any $r$, $γ_r$ denotes the $r^{th}$ norm of a gaussian, and $p^*$ is the dual norm of $p$.
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