We consider the problem of computing the q->p norm of a matrix A, which is defined for p,q \ge 1, as |A|_{q->p} = max_{x !=0 } |Ax|_p / |x|_q. This is in general a non-convex optimization problem, and is a natural generalization of the well-studied question of computing singular values (this corresponds to p=q=2). Different settings of parameters give rise to a variety of known interesting problems (such as the Grothendieck problem when p=1 and q=\infty). However, very little is understood about the approximability of the problem for different values of p,q. Our first result is an efficient algorithm for computing the q->p norm of matrices with non-negative entries, when q \ge p \ge 1. The algorithm we analyze is based on a natural fixed point iteration, which can be seen as an analog of power iteration for computing eigenvalues. We then present an application of our techniques to the problem of constructing a scheme for oblivious routing in the l_p norm. This makes constructive a recent existential result of Englert and Räcke [ER] on O(log n)-competitive oblivious routing schemes (which they make constructive only for p=2). On the other hand, when we do not have any restrictions on the entries (such as non-negativity), we prove that the problem is NP-hard to approximate to any constant factor, for 2 < p \le q, and p \le q < 2 (these are precisely the ranges of p,q with p\le q, where constant factor approximations are not known). In this range, our techniques also show that if NP does not have quasi-polynomial time algorithms, the q->p cannot be approximated to a factor 2^{(log n)^{1-eps}}, for any \eps>0.
证明了n × n实矩阵A的积和式的对数和2n × 2n实对称矩阵A的积和式的对数在加法误差1 > n> 0的范围内可以用A的元中的一个0次多项式p（ln n - ln n）来逼近，只要A的元a_ij满足delta < a_ij < 1 for an arbitrarily small delta >0，预先确定.此外，多项式p可以在n^{O（ln n - ln n）}时间内计算。我们还改进了复矩阵和复张量A的多维积和式的逼近ln per A、ln haf A和近似的界。
Approximating Hamiltonian dynamics with the Nyström method
INRIA（法国国家信息与自动化研究所）；University College London（伦敦大学学院）；Rahko Ltd.（拉赫科有限公司）；University of Texas at Austin（得克萨斯大学奥斯汀分校）；University of Oxford（牛津大学）；IIT（意大利理工学院）
Approximating Nash Equilibria and Dense Subgraphs via an Approximate Version of Carathéodory's Theorem
我们提出的算法应用的近似版本的Carathéodory定理。该定理指出，给定$\mathbb{R}^d$中的一组向量$X$，对于$X$的凸包中的每个向量，存在一个$\varepsilon$ -闭（对于$2\leq p < \infty$，在$p$ -范数距离下）向量，该向量可以表示为$X$的至多$b$向量的凸组合，其中，约束$b$取决于$\varepsilon$和范数$p$，并且与维度$d$无关。这个定理可以通过实例化Maurey引理来推导，早期的参考文献可以在Pisier（1981）和Carl（1985）的工作中找到。然而，在本文中，我们提出了一个独立的证明这一结果。利用这个定理，我们建立了在一个具有$ n \times n$支付矩阵$A, B$的双矩阵博弈中，如果$A+B$的任何一列中的非零元素的个数至多为$s$，那么这个博弈的$\varepsilon$ -纳什均衡可以在时间上计算出$n^{O\left(\frac{\log s }{\varepsilon^2}\right)}$ .这，特别是，给我们一个多项式时间近似计划纳什均衡在游戏中固定列稀疏$s$。此外，对于任意双矩阵游戏-因为$s$最多可以是$n$ -我们算法的运行时间与Lipton，Markakis和Mehta（2003）获得的最著名的上限相匹配。近似Carathéodory定理也导致了一个加性近似算法的规范化den…
